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Theorem unipr 3662
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
Hypotheses
Ref Expression
unipr.1 𝐴 ∈ V
unipr.2 𝐵 ∈ V
Assertion
Ref Expression
unipr {𝐴, 𝐵} = (𝐴𝐵)

Proof of Theorem unipr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.43 1564 . . . 4 (∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
2 vex 2622 . . . . . . . 8 𝑦 ∈ V
32elpr 3462 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43anbi2i 445 . . . . . 6 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)))
5 andi 767 . . . . . 6 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
64, 5bitri 182 . . . . 5 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
76exbii 1541 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
8 unipr.1 . . . . . . 7 𝐴 ∈ V
98clel3 2750 . . . . . 6 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
10 exancom 1544 . . . . . 6 (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
119, 10bitri 182 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
12 unipr.2 . . . . . . 7 𝐵 ∈ V
1312clel3 2750 . . . . . 6 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
14 exancom 1544 . . . . . 6 (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1513, 14bitri 182 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1611, 15orbi12i 716 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
171, 7, 163bitr4ri 211 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}))
1817abbii 2203 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
19 df-un 3001 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
20 df-uni 3649 . 2 {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
2118, 19, 203eqtr4ri 2119 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wo 664   = wceq 1289  wex 1426  wcel 1438  {cab 2074  Vcvv 2619  cun 2995  {cpr 3442   cuni 3648
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649
This theorem is referenced by:  uniprg  3663  unisn  3664  uniop  4073  unex  4257  bj-unex  11467
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