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Theorem unipr 3619
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 1533 . . . 4 (∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
2 vex 2575 . . . . . . . 8 𝑦 ∈ V
32elpr 3421 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43anbi2i 438 . . . . . 6 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)))
5 andi 740 . . . . . 6 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
64, 5bitri 177 . . . . 5 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
76exbii 1510 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
8 unipr.1 . . . . . . 7 𝐴 ∈ V
98clel3 2699 . . . . . 6 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
10 exancom 1513 . . . . . 6 (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
119, 10bitri 177 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
12 unipr.2 . . . . . . 7 𝐵 ∈ V
1312clel3 2699 . . . . . 6 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
14 exancom 1513 . . . . . 6 (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1513, 14bitri 177 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1611, 15orbi12i 689 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
171, 7, 163bitr4ri 206 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}))
1817abbii 2167 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
19 df-un 2947 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
20 df-uni 3606 . 2 {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
2118, 19, 203eqtr4ri 2085 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wo 637   = wceq 1257  wex 1395  wcel 1407  {cab 2040  Vcvv 2572  cun 2940  {cpr 3401   cuni 3605
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-uni 3606
This theorem is referenced by:  uniprg  3620  unisn  3621  uniop  4017  unex  4201  bj-unex  10369
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