ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssct Unicode version

Theorem ssct 6641
Description: A subset of a set dominated by  om is dominated by  om. (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 6577 . . . 4  |-  ( B  ~<_  om  ->  B  e.  _V )
2 ssdomg 6602 . . . 4  |-  ( B  e.  _V  ->  ( A  C_  B  ->  A  ~<_  B ) )
31, 2syl 14 . . 3  |-  ( B  ~<_  om  ->  ( A  C_  B  ->  A  ~<_  B ) )
43impcom 124 . 2  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  B )
5 domtr 6609 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
64, 5sylancom 414 1  |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1448   _Vcvv 2641    C_ wss 3021   class class class wbr 3875   omcom 4442    ~<_ cdom 6563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-dom 6566
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator