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Theorem unipr 3585
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 1516 . . . 4 (y((x y y = A) (x y y = B)) ↔ (y(x y y = A) y(x y y = B)))
2 vex 2554 . . . . . . . 8 y V
32elpr 3385 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
43anbi2i 430 . . . . . 6 ((x y y {A, B}) ↔ (x y (y = A y = B)))
5 andi 730 . . . . . 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 1493 . . . 4 (y(x y y {A, B}) ↔ y((x y y = A) (x y y = B)))
8 unipr.1 . . . . . . 7 A V
98clel3 2673 . . . . . 6 (x Ay(y = A x y))
10 exancom 1496 . . . . . 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 2673 . . . . . 6 (x By(y = B x y))
14 exancom 1496 . . . . . 6 (y(y = B x y) ↔ y(x y y = B))
1513, 14bitri 173 . . . . 5 (x By(x y y = B))
1611, 15orbi12i 680 . . . 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 2150 . 2 {x ∣ (x A x B)} = {xy(x y y {A, B})}
19 df-un 2916 . 2 (AB) = {x ∣ (x A x B)}
20 df-uni 3572 . 2 {A, B} = {xy(x y y {A, B})}
2118, 19, 203eqtr4ri 2068 1 {A, B} = (AB)
Colors of variables: wff set class
Syntax hints:   wa 97   wo 628   = wceq 1242  wex 1378   wcel 1390  {cab 2023  Vcvv 2551  cun 2909  {cpr 3368   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  uniprg  3586  unisn  3587  uniop  3983  unex  4142  bj-unex  9304
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