Theorem ssnct 45015

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 9072	. . 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 480	. 2 ⊢ ((𝜑 ∧ 𝐵 ≼ ω) → ¬ 𝐴 ≼ ω)
6	3, 5	pm2.65da 816	1 ⊢ (𝜑 → ¬ 𝐵 ≼ ω)

Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3931   class class class wbr 5123  ωcom 7868   ≼ cdom 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-dom 8968

This theorem is referenced by:  iocnct  45486  iccnct  45487