Theorem ssct 9115

Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5334, ax-un 7783. (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 9063	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3900   class class class wbr 5108  ωcom 7911   ≼ cdom 9009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-dom 9013

This theorem is used by:  measvuni  34958  measiuns  34961  sxbrsigalem1  35028  ssnct  46367  fzct  46663  fzoct  46668  salexct  47617  opnvonmbllem2  47916