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Theorem ssct 27762
Description: Any subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
ssct  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )

Proof of Theorem ssct
StepHypRef Expression
1 ctex 27761 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
2 ssdomg 7554 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  B  ->  A  ~<_  B ) )
31, 2syl 16 . . 3  |-  ( B  ~<_  om  ->  ( A  C_  B  ->  A  ~<_  B ) )
43impcom 428 . 2  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  B )
5 domtr 7561 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
64, 5sylancom 665 1  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439   omcom 6673    ~<_ cdom 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-dom 7511
This theorem is referenced by:  measvuni  28422  measiuns  28425  sxbrsigalem1  28493
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