Theorem unieq 3618

Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
unieq	|- ( A = B -> U. A = U. B )

Proof of Theorem unieq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2551	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2197	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3611	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3611	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2139	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1285   {cab 2068   E.wrex 2350   U.cuni 3609

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-uni 3610

This theorem is referenced by:  unieqi  3619  unieqd  3620  uniintsnr  3680  iununir  3767  treq  3889  limeq  4140  uniex  4200  uniexg  4201  ordsucunielexmid  4282  onsucuni2  4315  elvvuni  4430  unielrel  4875  unixp0im  4884  iotass  4914  nnsucuniel  6139  en1bg  6347  bj-uniex  10893  bj-uniexg  10894