MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniop Structured version   Visualization version   GIF version

Theorem uniop 5370
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 4762 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 4813 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snex 5297 . . 3 {𝐴} ∈ V
6 prex 5298 . . 3 {𝐴, 𝐵} ∈ V
75, 6unipr 4817 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
8 snsspr1 4707 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
9 ssequn1 4107 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
108, 9mpbi 233 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
114, 7, 103eqtri 2825 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2111  Vcvv 3441  cun 3879  wss 3881  {csn 4525  {cpr 4527  cop 4531   cuni 4800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801
This theorem is referenced by:  uniopel  5371  elvvuni  5592  dmrnssfld  5806  dffv2  6733  rankxplim  9292
  Copyright terms: Public domain W3C validator