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Theorem unipr 4843
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 1874 . . . 4 (∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
2 vex 3495 . . . . . . . 8 𝑦 ∈ V
32elpr 4580 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
43anbi2i 622 . . . . . 6 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)))
5 andi 1001 . . . . . 6 ((𝑥𝑦 ∧ (𝑦 = 𝐴𝑦 = 𝐵)) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
64, 5bitri 276 . . . . 5 ((𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
76exbii 1839 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑥𝑦𝑦 = 𝐴) ∨ (𝑥𝑦𝑦 = 𝐵)))
8 unipr.1 . . . . . . 7 𝐴 ∈ V
98clel3 3652 . . . . . 6 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
10 exancom 1852 . . . . . 6 (∃𝑦(𝑦 = 𝐴𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
119, 10bitri 276 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐴))
12 unipr.2 . . . . . . 7 𝐵 ∈ V
1312clel3 3652 . . . . . 6 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
14 exancom 1852 . . . . . 6 (∃𝑦(𝑦 = 𝐵𝑥𝑦) ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1513, 14bitri 276 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑥𝑦𝑦 = 𝐵))
1611, 15orbi12i 908 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (∃𝑦(𝑥𝑦𝑦 = 𝐴) ∨ ∃𝑦(𝑥𝑦𝑦 = 𝐵)))
171, 7, 163bitr4ri 305 . . 3 ((𝑥𝐴𝑥𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵}))
1817abbii 2883 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
19 df-un 3938 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
20 df-uni 4831 . 2 {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦 ∈ {𝐴, 𝐵})}
2118, 19, 203eqtr4ri 2852 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  {cab 2796  Vcvv 3492  cun 3931  {cpr 4559   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560  df-uni 4831
This theorem is referenced by:  uniprg  4844  uniintsn  4904  uniop  5396  unex  7458  rankxplim  9296  mrcun  16881  indistps  21547  indistps2  21548  leordtval2  21748  ex-uni  28132  mnuprdlem1  40485  mnuprdlem2  40486  mnurndlem1  40494  fouriersw  42393
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