MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl5eqssr Unicode version

Theorem syl5eqssr 3223
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqssr.1  |-  B  =  A
syl5eqssr.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
syl5eqssr  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl5eqssr
StepHypRef Expression
1 syl5eqssr.1 . . 3  |-  B  =  A
21eqcomi 2287 . 2  |-  A  =  B
3 syl5eqssr.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3syl5eqss 3222 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152
This theorem is referenced by:  relcnvtr  5192  fimacnvdisj  5419  dffv2  5592  fimacnv  5657  f1ompt  5682  fnwelem  6230  tfrlem15  6408  omxpenlem  6963  hartogslem1  7257  infxpidm2  7644  alephgeom  7709  infenaleph  7718  cfflb  7885  pwfseqlem5  8285  imasvscafn  13439  mrieqvlemd  13531  cnvps  14321  dirdm  14356  tsrdir  14360  frmdss2  14485  iinopn  16648  xkococnlem  17353  tgpconcomp  17795  mbfconstlem  18984  itg2seq  19097  limcdif  19226  dvres2lem  19260  c1lip3  19346  lhop  19363  plyeq0  19593  dchrghm  20495  chssoc  22075  cvmliftmolem1  23812  cvmlift2lem9a  23834  cvmlift2lem9  23842  cartarlim  25905  pgapspf  26052  cnres2  26483  rngunsnply  27378  proot1hash  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
  Copyright terms: Public domain W3C validator