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Theorem uniop 5506
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 4865 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 4912 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5422 . . 3 {𝐴} ∈ V
6 prex 5423 . . 3 {𝐴, 𝐵} ∈ V
75, 6unipr 4917 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
8 snsspr1 4810 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
9 ssequn1 4173 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
108, 9mpbi 229 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
114, 7, 103eqtri 2756 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  cun 3939  wss 3941  {csn 4621  {cpr 4623  cop 4627   cuni 4900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901
This theorem is referenced by:  uniopel  5507  elvvuni  5743  dmrnssfld  5960  dffv2  6977  rankxplim  9871
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