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Theorem opeqsn 4444
 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 3975 . . 3
43eqeq1i 2442 . 2
5 snex 4397 . . 3
6 prex 4398 . . 3
7 opeqsn.3 . . 3
85, 6, 7preqsn 3972 . 2
9 eqcom 2437 . . . . 5
101, 2, 1preqsn 3972 . . . . 5
11 eqcom 2437 . . . . . . 7
1211anbi2i 676 . . . . . 6
13 anidm 626 . . . . . 6
1412, 13bitri 241 . . . . 5
159, 10, 143bitri 263 . . . 4
1615anbi1i 677 . . 3
17 dfsn2 3820 . . . . . . 7
18 preq2 3876 . . . . . . 7
1917, 18syl5req 2480 . . . . . 6
2019eqeq1d 2443 . . . . 5
21 eqcom 2437 . . . . 5
2220, 21syl6bb 253 . . . 4
2322pm5.32i 619 . . 3
2416, 23bitri 241 . 2
254, 8, 243bitri 263 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948  csn 3806  cpr 3807  cop 3809 This theorem is referenced by:  relop  5015 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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