Theorem ssnct 44068

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	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssnct	⊢ (𝜑 → ¬ 𝐵 ≼ ω)

Proof of Theorem ssnct

Step	Hyp	Ref	Expression
1		ssnct.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssct 9053	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
3	1, 2	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
4		ssnct.1	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ ω)
5	4	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
6	3, 5	pm2.65da 815	1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3948   class class class wbr 5148  ωcom 7857   ≼ cdom 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-dom 8943

This theorem is referenced by:  iocnct  44552  iccnct  44553