Theorem unieq 3858

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 → ∪ 𝐴 = ∪ 𝐵)

Proof of Theorem unieq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2702	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
2	1	abbidv 2322	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥})
3		dfuni2 3851	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
4		dfuni2 3851	. 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
5	2, 3, 4	3eqtr4g 2262	1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1372  {cab 2190  ∃wrex 2484  ∪ cuni 3849

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-uni 3850

This theorem is referenced by:  unieqi  3859  unieqd  3860  uniintsnr  3920  iununir  4010  treq  4147  limeq  4423  uniex  4483  uniexg  4485  ordsucunielexmid  4578  onsucuni2  4611  nnpredcl  4670  elvvuni  4738  unielrel  5209  unixp0im  5218  iotass  5248  nnsucuniel  6580  en1bg  6891  omp1eom  7196  ctmlemr  7209  nnnninfeq2  7230  uniopn  14444  istopon  14456  eltg3  14500  tgdom  14515  cldval  14542  ntrfval  14543  clsfval  14544  neifval  14583  tgrest  14612  cnprcl2k  14649  bj-uniex  15815  bj-uniexg  15816  nnsf  15904  peano3nninf  15906  exmidsbthr  15924