Theorem ssct 8966

Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5298, ax-un 7663. (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 8915	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3897   class class class wbr 5086  ωcom 7791   ≼ cdom 8862

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-dom 8866

This theorem is referenced by:  measvuni  34219  measiuns  34222  sxbrsigalem1  34290  ssnct  45114  fzct  45417  fzoct  45422  salexct  46372  opnvonmbllem2  46671