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Theorem uniop 4184
 Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1
opthw.2
Assertion
Ref Expression
uniop

Proof of Theorem uniop
StepHypRef Expression
1 opthw.1 . . . 4
2 opthw.2 . . . 4
31, 2dfop 3711 . . 3
43unieqi 3753 . 2
51snex 4116 . . 3
6 prexg 4140 . . . 4
71, 2, 6mp2an 423 . . 3
85, 7unipr 3757 . 2
9 snsspr1 3675 . . 3
10 ssequn1 3250 . . 3
119, 10mpbi 144 . 2
124, 8, 113eqtri 2165 1
 Colors of variables: wff set class Syntax hints:   wceq 1332   wcel 1481  cvv 2689   cun 3073   wss 3075  csn 3531  cpr 3532  cop 3534  cuni 3743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744 This theorem is referenced by:  uniopel  4185  elvvuni  4610  dmrnssfld  4809
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