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Theorem uniop 4019
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 3575 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3617 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snexgOLD 3962 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
61, 5ax-mp 7 . . 3 {𝐴} ∈ V
7 prexgOLD 3973 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
81, 2, 7mp2an 410 . . 3 {𝐴, 𝐵} ∈ V
96, 8unipr 3621 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
10 snsspr1 3539 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
11 ssequn1 3140 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
1210, 11mpbi 137 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
134, 9, 123eqtri 2080 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cun 2942  wss 2944  {csn 3402  {cpr 3403  cop 3405   cuni 3607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608
This theorem is referenced by:  uniopel  4020  elvvuni  4431  dmrnssfld  4622
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