Theorem unieq 3897

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 2729	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2347	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3890	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3890	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1395   {cab 2215   E.wrex 2509   U.cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3889

This theorem is referenced by:  unieqi  3898  unieqd  3899  uniintsnr  3959  iununir  4049  treq  4188  limeq  4468  uniex  4528  uniexg  4530  ordsucunielexmid  4623  onsucuni2  4656  nnpredcl  4715  elvvuni  4783  unielrel  5256  unixp0im  5265  iotass  5296  nnsucuniel  6641  en1bg  6952  omp1eom  7262  ctmlemr  7275  nnnninfeq2  7296  uniopn  14675  istopon  14687  eltg3  14731  tgdom  14746  cldval  14773  ntrfval  14774  clsfval  14775  neifval  14814  tgrest  14843  cnprcl2k  14880  bj-uniex  16280  bj-uniexg  16281  nnsf  16371  peano3nninf  16373  exmidsbthr  16391