Theorem unieq 3745

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 2627	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2257	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3738	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3738	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2125   E.wrex 2417   U.cuni 3736

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-uni 3737

This theorem is referenced by:  unieqi  3746  unieqd  3747  uniintsnr  3807  iununir  3896  treq  4032  limeq  4299  uniex  4359  uniexg  4361  ordsucunielexmid  4446  onsucuni2  4479  nnpredcl  4536  elvvuni  4603  unielrel  5066  unixp0im  5075  iotass  5105  nnsucuniel  6391  en1bg  6694  omp1eom  6980  ctmlemr  6993  uniopn  12168  istopon  12180  eltg3  12226  tgdom  12241  cldval  12268  ntrfval  12269  clsfval  12270  neifval  12309  tgrest  12338  cnprcl2k  12375  bj-uniex  13115  bj-uniexg  13116  nnsf  13199  peano3nninf  13201  exmidsbthr  13218