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Theorem ssnct 45030
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 9096 . . 3 ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
31, 2sylan 580 . 2 ((𝜑𝐵 ≼ ω) → 𝐴 ≼ ω)
4 ssnct.1 . . 3 (𝜑 → ¬ 𝐴 ≼ ω)
54adantr 480 . 2 ((𝜑𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
63, 5pm2.65da 817 1 (𝜑 → ¬ 𝐵 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wss 3964   class class class wbr 5149  ωcom 7891  cdom 8988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5150  df-opab 5212  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-dom 8992
This theorem is referenced by:  iocnct  45505  iccnct  45506
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