Theorem ssct 9048

Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5354, ax-un 7719. (Revised by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
ssct	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)

Proof of Theorem ssct

Step	Hyp	Ref	Expression
1		domssl 8991	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3941   class class class wbr 5139  ωcom 7849   ≼ cdom 8934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-dom 8938

This theorem is referenced by:  measvuni  33704  measiuns  33707  sxbrsigalem1  33776  ssnct  44279  fzct  44599  fzoct  44604  salexct  45560  opnvonmbllem2  45859