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Theorem uniop 4177
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 3704 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3746 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 4109 . . 3 {𝐴} ∈ V
6 prexg 4133 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 422 . . 3 {𝐴, 𝐵} ∈ V
85, 7unipr 3750 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9 snsspr1 3668 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
10 ssequn1 3246 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
119, 10mpbi 144 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
124, 8, 113eqtri 2164 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  Vcvv 2686  cun 3069  wss 3071  {csn 3527  {cpr 3528  cop 3530   cuni 3736
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737
This theorem is referenced by:  uniopel  4178  elvvuni  4603  dmrnssfld  4802
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