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Theorem uniop 3992
 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 3548 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3590 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
5 snexgOLD 3935 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
61, 5ax-mp 7 . . 3 {𝐴} ∈ V
7 prexgOLD 3946 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
81, 2, 7mp2an 402 . . 3 {𝐴, 𝐵} ∈ V
96, 8unipr 3594 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
10 snsspr1 3512 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
11 ssequn1 3113 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
1210, 11mpbi 133 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
134, 9, 123eqtri 2064 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  {csn 3375  {cpr 3376  ⟨cop 3378  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581 This theorem is referenced by:  uniopel  3993  elvvuni  4404  dmrnssfld  4595
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