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Theorem ssct 8284
Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
ssct ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)

Proof of Theorem ssct
StepHypRef Expression
1 ctex 8211 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
2 ssdomg 8242 . . . 4 (𝐵 ∈ V → (𝐴𝐵𝐴𝐵))
31, 2syl 17 . . 3 (𝐵 ≼ ω → (𝐴𝐵𝐴𝐵))
43impcom 397 . 2 ((𝐴𝐵𝐵 ≼ ω) → 𝐴𝐵)
5 domtr 8249 . 2 ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
64, 5sylancom 583 1 ((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  Vcvv 3386  wss 3770   class class class wbr 4844  ωcom 7300  cdom 8194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-dom 8198
This theorem is referenced by:  measvuni  30792  measiuns  30795  sxbrsigalem1  30862  ssnct  40003  fzct  40335  fzoct  40342  salexct  41290  opnvonmbllem2  41588
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