Theorem unieq 3859

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 2703	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2323	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3852	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3852	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2191   E.wrex 2485   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-uni 3851

This theorem is referenced by:  unieqi  3860  unieqd  3861  uniintsnr  3921  iununir  4011  treq  4149  limeq  4425  uniex  4485  uniexg  4487  ordsucunielexmid  4580  onsucuni2  4613  nnpredcl  4672  elvvuni  4740  unielrel  5211  unixp0im  5220  iotass  5250  nnsucuniel  6583  en1bg  6894  omp1eom  7199  ctmlemr  7212  nnnninfeq2  7233  uniopn  14506  istopon  14518  eltg3  14562  tgdom  14577  cldval  14604  ntrfval  14605  clsfval  14606  neifval  14645  tgrest  14674  cnprcl2k  14711  bj-uniex  15890  bj-uniexg  15891  nnsf  15979  peano3nninf  15981  exmidsbthr  15999