Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssct Structured version   Unicode version

Theorem ssct 26145
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 26144 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
2 ssdomg 7457 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  B  ->  A  ~<_  B ) )
31, 2syl 16 . . 3  |-  ( B  ~<_  om  ->  ( A  C_  B  ->  A  ~<_  B ) )
43impcom 430 . 2  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  B )
5 domtr 7464 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
64, 5sylancom 667 1  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   _Vcvv 3070    C_ wss 3428   class class class wbr 4392   omcom 6578    ~<_ cdom 7410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-dom 7414
This theorem is referenced by:  measvuni  26764  measiuns  26767  sxbrsigalem1  26836
  Copyright terms: Public domain W3C validator