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Theorem uniop 3983
 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 3539 . . 3 A, B⟩ = {{A}, {A, B}}
43unieqi 3581 . 2 A, B⟩ = {{A}, {A, B}}
5 snexgOLD 3926 . . . 4 (A V → {A} V)
61, 5ax-mp 7 . . 3 {A} V
7 prexgOLD 3937 . . . 4 ((A V B V) → {A, B} V)
81, 2, 7mp2an 402 . . 3 {A, B} V
96, 8unipr 3585 . 2 {{A}, {A, B}} = ({A} ∪ {A, B})
10 snsspr1 3503 . . 3 {A} ⊆ {A, B}
11 ssequn1 3107 . . 3 ({A} ⊆ {A, B} ↔ ({A} ∪ {A, B}) = {A, B})
1210, 11mpbi 133 . 2 ({A} ∪ {A, B}) = {A, B}
134, 9, 123eqtri 2061 1 A, B⟩ = {A, B}
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {csn 3367  {cpr 3368  ⟨cop 3370  ∪ cuni 3571 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572 This theorem is referenced by:  uniopel  3984  elvvuni  4347  dmrnssfld  4538
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