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Theorem uniop 4054
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 3603 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3645 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 3992 . . 3 {𝐴} ∈ V
6 prexg 4010 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 417 . . 3 {𝐴, 𝐵} ∈ V
85, 7unipr 3649 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9 snsspr1 3567 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
10 ssequn1 3159 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
119, 10mpbi 143 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
124, 8, 113eqtri 2109 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  Vcvv 2615  cun 2986  wss 2988  {csn 3430  {cpr 3431  cop 3433   cuni 3635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3930  ax-pow 3982  ax-pr 4008
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3416  df-sn 3436  df-pr 3437  df-op 3439  df-uni 3636
This theorem is referenced by:  uniopel  4055  elvvuni  4468  dmrnssfld  4662
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