Theorem unieq 3753

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 2630	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2258	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3746	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3746	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2198	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1332   {cab 2126   E.wrex 2418   U.cuni 3744

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-uni 3745

This theorem is referenced by:  unieqi  3754  unieqd  3755  uniintsnr  3815  iununir  3904  treq  4040  limeq  4307  uniex  4367  uniexg  4369  ordsucunielexmid  4454  onsucuni2  4487  nnpredcl  4544  elvvuni  4611  unielrel  5074  unixp0im  5083  iotass  5113  nnsucuniel  6399  en1bg  6702  omp1eom  6988  ctmlemr  7001  uniopn  12207  istopon  12219  eltg3  12265  tgdom  12280  cldval  12307  ntrfval  12308  clsfval  12309  neifval  12348  tgrest  12377  cnprcl2k  12414  bj-uniex  13286  bj-uniexg  13287  nnsf  13374  peano3nninf  13376  exmidsbthr  13393