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

Theorem opid 4004
 Description: The ordered pair in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
Hypothesis
Ref Expression
opid.1
Assertion
Ref Expression
opid

Proof of Theorem opid
StepHypRef Expression
1 dfsn2 3830 . . . 4
21eqcomi 2442 . . 3
32preq2i 3889 . 2
4 opid.1 . . 3
54, 4dfop 3985 . 2
6 dfsn2 3830 . 2
73, 5, 63eqtr4i 2468 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  cvv 2958  csn 3816  cpr 3817  cop 3819 This theorem is referenced by:  dmsnsnsn  5350  funopg  5487 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 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
 Copyright terms: Public domain W3C validator