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Theorem unipr 3568
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 1501 . . . 4 (y((x y y = A) (x y y = B)) ↔ (y(x y y = A) y(x y y = B)))
2 vex 2538 . . . . . . . 8 y V
32elpr 3368 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
43anbi2i 433 . . . . . 6 ((x y y {A, B}) ↔ (x y (y = A y = B)))
5 andi 719 . . . . . 6 ((x y (y = A y = B)) ↔ ((x y y = A) (x y y = B)))
64, 5bitri 173 . . . . 5 ((x y y {A, B}) ↔ ((x y y = A) (x y y = B)))
76exbii 1478 . . . 4 (y(x y y {A, B}) ↔ y((x y y = A) (x y y = B)))
8 unipr.1 . . . . . . 7 A V
98clel3 2656 . . . . . 6 (x Ay(y = A x y))
10 exancom 1481 . . . . . 6 (y(y = A x y) ↔ y(x y y = A))
119, 10bitri 173 . . . . 5 (x Ay(x y y = A))
12 unipr.2 . . . . . . 7 B V
1312clel3 2656 . . . . . 6 (x By(y = B x y))
14 exancom 1481 . . . . . 6 (y(y = B x y) ↔ y(x y y = B))
1513, 14bitri 173 . . . . 5 (x By(x y y = B))
1611, 15orbi12i 668 . . . 4 ((x A x B) ↔ (y(x y y = A) y(x y y = B)))
171, 7, 163bitr4ri 202 . . 3 ((x A x B) ↔ y(x y y {A, B}))
1817abbii 2135 . 2 {x ∣ (x A x B)} = {xy(x y y {A, B})}
19 df-un 2899 . 2 (AB) = {x ∣ (x A x B)}
20 df-uni 3555 . 2 {A, B} = {xy(x y y {A, B})}
2118, 19, 203eqtr4ri 2053 1 {A, B} = (AB)
Colors of variables: wff set class
Syntax hints:   wa 97   wo 616   = wceq 1228  wex 1362   wcel 1374  {cab 2008  Vcvv 2535  cun 2892  {cpr 3351   cuni 3554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555
This theorem is referenced by:  uniprg  3569  unisn  3570  uniop  3966  unex  4126  bj-unex  7142
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