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Theorem unipr 3718
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 1590 . . . 4 (∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
2 vex 2661 . . . . . . . 8 𝑦 ∈ V
32elpr 3516 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43anbi2i 450 . . . . . 6 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)))
5 andi 790 . . . . . 6 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
64, 5bitri 183 . . . . 5 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
76exbii 1567 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
8 unipr.1 . . . . . . 7 𝐴 ∈ V
98clel3 2792 . . . . . 6 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
10 exancom 1570 . . . . . 6 (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
119, 10bitri 183 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
12 unipr.2 . . . . . . 7 𝐵 ∈ V
1312clel3 2792 . . . . . 6 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
14 exancom 1570 . . . . . 6 (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1513, 14bitri 183 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1611, 15orbi12i 736 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
171, 7, 163bitr4ri 212 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}))
1817abbii 2231 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
19 df-un 3043 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
20 df-uni 3705 . 2 {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
2118, 19, 203eqtr4ri 2147 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wo 680   = wceq 1314  wex 1451  wcel 1463  {cab 2101  Vcvv 2658  cun 3037  {cpr 3496   cuni 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-uni 3705
This theorem is referenced by:  uniprg  3719  unisn  3720  uniop  4145  unex  4330  bj-unex  12919
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