Theorem unieq 3836

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 2687	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2307	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3829	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3829	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2247	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2175   E.wrex 2469   U.cuni 3827

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-uni 3828

This theorem is referenced by:  unieqi  3837  unieqd  3838  uniintsnr  3898  iununir  3988  treq  4125  limeq  4398  uniex  4458  uniexg  4460  ordsucunielexmid  4551  onsucuni2  4584  nnpredcl  4643  elvvuni  4711  unielrel  5177  unixp0im  5186  iotass  5216  nnsucuniel  6524  en1bg  6830  omp1eom  7128  ctmlemr  7141  nnnninfeq2  7162  uniopn  13986  istopon  13998  eltg3  14042  tgdom  14057  cldval  14084  ntrfval  14085  clsfval  14086  neifval  14125  tgrest  14154  cnprcl2k  14191  bj-uniex  15155  bj-uniexg  15156  nnsf  15241  peano3nninf  15243  exmidsbthr  15259