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Theorem uniop 3966
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 A V
opthw.2 B V
Assertion
Ref Expression
uniop A, B⟩ = {A, B}

Proof of Theorem uniop
StepHypRef Expression
1 opthw.1 . . . 4 A V
2 opthw.2 . . . 4 B V
31, 2dfop 3522 . . 3 A, B⟩ = {{A}, {A, B}}
43unieqi 3564 . 2 A, B⟩ = {{A}, {A, B}}
5 snexgOLD 3909 . . . 4 (A V → {A} V)
61, 5ax-mp 7 . . 3 {A} V
7 prexgOLD 3920 . . . 4 ((A V B V) → {A, B} V)
81, 2, 7mp2an 404 . . 3 {A, B} V
96, 8unipr 3568 . 2 {{A}, {A, B}} = ({A} ∪ {A, B})
10 snsspr1 3486 . . 3 {A} ⊆ {A, B}
11 ssequn1 3090 . . 3 ({A} ⊆ {A, B} ↔ ({A} ∪ {A, B}) = {A, B})
1210, 11mpbi 133 . 2 ({A} ∪ {A, B}) = {A, B}
134, 9, 123eqtri 2046 1 A, B⟩ = {A, B}
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  cun 2892  wss 2894  {csn 3350  {cpr 3351  cop 3353   cuni 3554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555
This theorem is referenced by:  uniopel  3967  elvvuni  4331  dmrnssfld  4522
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