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Theorem ssnct 43637
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 9039 . . 3 ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
31, 2sylan 581 . 2 ((𝜑𝐵 ≼ ω) → 𝐴 ≼ ω)
4 ssnct.1 . . 3 (𝜑 → ¬ 𝐴 ≼ ω)
54adantr 482 . 2 ((𝜑𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
63, 5pm2.65da 816 1 (𝜑 → ¬ 𝐵 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wss 3946   class class class wbr 5144  ωcom 7842  cdom 8925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-dom 8929
This theorem is referenced by:  iocnct  44126  iccnct  44127
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