Theorem unieq 3783

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 2653	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2275	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3776	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3776	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2215	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1335   {cab 2143   E.wrex 2436   U.cuni 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-uni 3775

This theorem is referenced by:  unieqi  3784  unieqd  3785  uniintsnr  3845  iununir  3934  treq  4070  limeq  4339  uniex  4399  uniexg  4401  ordsucunielexmid  4492  onsucuni2  4525  nnpredcl  4584  elvvuni  4652  unielrel  5115  unixp0im  5124  iotass  5154  nnsucuniel  6444  en1bg  6747  omp1eom  7041  ctmlemr  7054  nnnninfeq2  7074  uniopn  12469  istopon  12481  eltg3  12527  tgdom  12542  cldval  12569  ntrfval  12570  clsfval  12571  neifval  12610  tgrest  12639  cnprcl2k  12676  bj-uniex  13563  bj-uniexg  13564  nnsf  13648  peano3nninf  13650  exmidsbthr  13665