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Theorem uniop 4147
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 3674 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3716 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 4079 . . 3 {𝐴} ∈ V
6 prexg 4103 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 422 . . 3 {𝐴, 𝐵} ∈ V
85, 7unipr 3720 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9 snsspr1 3638 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
10 ssequn1 3216 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
119, 10mpbi 144 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
124, 8, 113eqtri 2142 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  Vcvv 2660  cun 3039  wss 3041  {csn 3497  {cpr 3498  cop 3500   cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707
This theorem is referenced by:  uniopel  4148  elvvuni  4573  dmrnssfld  4772
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