EMB-505 captain reported inoperative anti-ice systems with bleeds 1 and 2 off CAS messages during taxi out. During climb; the cabin did not pressurize and the flight diverted to a safe landing.

Narrative

On the prior flight; during taxi-in; we experienced a BLEED 1 OFF and BLEED 2 OFF CAS message. We contacted Maintenance to inquire about any info regarding this condition; and they suggested that it may have been due to a low power setting during taxi. This led us to assume that the BLEED OFF messages were a non-issue that may resolve itself during the next takeoff. During the taxi-out for the next flight; the outside temperature was slightly above freezing; and the BLEED 1 OFF and BLEED 2 OFF messages persisted. We tested the anti-ice equipment of the aircraft; and it appeared that the Engine A/I systems worked and the WingStab A/I was inhibited. I assumed; based on the prior conversation with maintenance; that the inhibited A/I was due to the BLEED OFF messages which were in turn due to a low power setting and that it all would work normally once we applied power to commence takeoff. When we applied power for takeoff and the messages did not immediately resolve themselves; I assumed it was due to K-coding and that they would resolve shortly after takeoff. When we began climbing after takeoff; I could tell the cabin was not pressurizing; and shortly after takeoff; we began accumulating ice while the WINGSTAB A/I INHIBIT message was displayed. As we were climbing; the CABIN ALTITUDE HI warning CAS displayed. We leveled off the aircraft at approximately 10;000 ft; [requested assistance] and diverted to ZZZ due to its slightly above freezing temperature and that the route would remove us from icing conditions. While enroute; we elected to not do a CABIN ALTITUDE HI QRH due to our relatively low altitude and the fact that donning masks and dropping the pax masks would heighten the stress of the situation. We decided that the cabin altitude; while high; was safe without further action. The remainder of the flight was relatively uneventful. We soon exited icing conditions; calculated landing performance information for ZZZ; and landed normally with no further assistance from ATC.Suggestions: Gain a better understanding of the aircraft systems and refrain from taking off when we don't fully understand why the systems are behaving abnormally.