Theorem unieq 3907

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 2732	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2350	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3900	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3900	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2217   E.wrex 2512   U.cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899

This theorem is referenced by:  unieqi  3908  unieqd  3909  uniintsnr  3969  iununir  4059  treq  4198  limeq  4480  uniex  4540  uniexg  4542  ordsucunielexmid  4635  onsucuni2  4668  nnpredcl  4727  elvvuni  4796  unielrel  5271  unixp0im  5280  iotass  5311  nnsucuniel  6706  en1bg  7017  omp1eom  7354  ctmlemr  7367  nnnninfeq2  7388  uniopn  14812  istopon  14824  eltg3  14868  tgdom  14883  cldval  14910  ntrfval  14911  clsfval  14912  neifval  14951  tgrest  14980  cnprcl2k  15017  bj-uniex  16633  bj-uniexg  16634  nnsf  16731  peano3nninf  16733  exmidsbthr  16751