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Theorem uniop 5006
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 4432 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 4477 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 4938 . . 3 {𝐴} ∈ V
6 prex 4939 . . 3 {𝐴, 𝐵} ∈ V
75, 6unipr 4481 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
8 snsspr1 4377 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
9 ssequn1 3816 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
108, 9mpbi 220 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
114, 7, 103eqtri 2677 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  {csn 4210  {cpr 4212  cop 4216   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469
This theorem is referenced by:  uniopel  5007  elvvuni  5213  dmrnssfld  5416  dffv2  6310  rankxplim  8780
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