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

Theorem syl5eqssr 3394
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 2441 . 2  |-  A  =  B
3 syl5eqssr.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3syl5eqss 3393 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3321
This theorem is referenced by:  relcnvtr  5390  fimacnvdisj  5622  dffv2  5797  fimacnv  5863  f1ompt  5892  fnwelem  6462  tfrlem15  6654  omxpenlem  7210  hartogslem1  7512  infxpidm2  7899  alephgeom  7964  infenaleph  7973  cfflb  8140  pwfseqlem5  8539  imasvscafn  13763  mrieqvlemd  13855  cnvps  14645  dirdm  14680  tsrdir  14684  frmdss2  14809  iinopn  16976  neitr  17245  xkococnlem  17692  tgpconcomp  18143  trcfilu  18325  mbfconstlem  19522  itg2seq  19635  limcdif  19764  dvres2lem  19798  c1lip3  19884  lhop  19901  plyeq0  20131  dchrghm  21041  chssoc  22999  hauseqcn  24294  cvmliftmolem1  24969  cvmlift2lem9a  24991  cvmlift2lem9  24999  cnres2  26473  rngunsnply  27356  proot1hash  27497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-in 3328  df-ss 3335
  Copyright terms: Public domain W3C validator