Theorem unieq 3848

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 2694	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2314	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3841	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3841	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2182   E.wrex 2476   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-uni 3840

This theorem is referenced by:  unieqi  3849  unieqd  3850  uniintsnr  3910  iununir  4000  treq  4137  limeq  4412  uniex  4472  uniexg  4474  ordsucunielexmid  4567  onsucuni2  4600  nnpredcl  4659  elvvuni  4727  unielrel  5197  unixp0im  5206  iotass  5236  nnsucuniel  6553  en1bg  6859  omp1eom  7161  ctmlemr  7174  nnnninfeq2  7195  uniopn  14237  istopon  14249  eltg3  14293  tgdom  14308  cldval  14335  ntrfval  14336  clsfval  14337  neifval  14376  tgrest  14405  cnprcl2k  14442  bj-uniex  15563  bj-uniexg  15564  nnsf  15649  peano3nninf  15651  exmidsbthr  15667