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

Theorem opeqsn 4697
 Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqsn.1
opeqsn.2
opeqsn.3
Assertion
Ref Expression
opeqsn

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . . . 4
2 opeqsn.2 . . . 4
31, 2dfop 4157 . . 3
43eqeq1i 2476 . 2
5 snex 4641 . . 3
6 prex 4642 . . 3
7 opeqsn.3 . . 3
85, 6, 7preqsn 4151 . 2
9 eqcom 2478 . . . . 5
101, 2, 1preqsn 4151 . . . . 5
11 eqcom 2478 . . . . . . 7
1211anbi2i 708 . . . . . 6
13 anidm 656 . . . . . 6
1412, 13bitri 257 . . . . 5
159, 10, 143bitri 279 . . . 4
1615anbi1i 709 . . 3
17 dfsn2 3972 . . . . . . 7
18 preq2 4043 . . . . . . 7
1917, 18syl5req 2518 . . . . . 6
2019eqeq1d 2473 . . . . 5
21 eqcom 2478 . . . . 5
2220, 21syl6bb 269 . . . 4
2322pm5.32i 649 . . 3
2416, 23bitri 257 . 2
254, 8, 243bitri 279 1
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   wceq 1452   wcel 1904  cvv 3031  csn 3959  cpr 3961  cop 3965 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966 This theorem is referenced by:  relop  4990  snopeqop  39145  propeqop  39146
 Copyright terms: Public domain W3C validator