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Theorem ssnct 39746
Description: A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
ssnct.1 (𝜑 → ¬ 𝐴 ≼ ω)
ssnct.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssnct (𝜑 → ¬ 𝐵 ≼ ω)

Proof of Theorem ssnct
StepHypRef Expression
1 ssnct.2 . . 3 (𝜑𝐴𝐵)
2 ssct 8204 . . 3 ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
31, 2sylan 489 . 2 ((𝜑𝐵 ≼ ω) → 𝐴 ≼ ω)
4 ssnct.1 . . 3 (𝜑 → ¬ 𝐴 ≼ ω)
54adantr 472 . 2 ((𝜑𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
63, 5pm2.65da 601 1 (𝜑 → ¬ 𝐵 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wss 3713   class class class wbr 4802  ωcom 7228  cdom 8117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-dom 8121
This theorem is referenced by:  iocnct  40268  iccnct  40269
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