Theorem ssnct 41348

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 8600	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
3	1, 2	sylan 582	. 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
4		ssnct.1	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ ω)
5	4	adantr 483	. 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
6	3, 5	pm2.65da 815	1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3938   class class class wbr 5068  ωcom 7582   ≼ cdom 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-dom 8513

This theorem is referenced by:  iocnct  41823  iccnct  41824