Theorem ssnct 45198

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 8978	. . 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 3898   class class class wbr 5093  ωcom 7802   ≼ cdom 8873

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-dom 8877

This theorem is referenced by:  iocnct  45664  iccnct  45665