Theorem uniprg 3569

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)

Assertion

Ref	Expression
uniprg	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ {A, B} = (A ∪ B))

Proof of Theorem uniprg

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

Step	Hyp	Ref	Expression
1		preq1 3421	. . . 4 ⊢ (x = A → {x, y} = {A, y})
2	1	unieqd 3565	. . 3 ⊢ (x = A → ∪ {x, y} = ∪ {A, y})
3		uneq1 3067	. . 3 ⊢ (x = A → (x ∪ y) = (A ∪ y))
4	2, 3	eqeq12d 2036	. 2 ⊢ (x = A → (∪ {x, y} = (x ∪ y) ↔ ∪ {A, y} = (A ∪ y)))
5		preq2 3422	. . . 4 ⊢ (y = B → {A, y} = {A, B})
6	5	unieqd 3565	. . 3 ⊢ (y = B → ∪ {A, y} = ∪ {A, B})
7		uneq2 3068	. . 3 ⊢ (y = B → (A ∪ y) = (A ∪ B))
8	6, 7	eqeq12d 2036	. 2 ⊢ (y = B → (∪ {A, y} = (A ∪ y) ↔ ∪ {A, B} = (A ∪ B)))
9		vex 2538	. . 3 ⊢ x ∈ V
10		vex 2538	. . 3 ⊢ y ∈ V
11	9, 10	unipr 3568	. 2 ⊢ ∪ {x, y} = (x ∪ y)
12	4, 8, 11	vtocl2g 2594	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ {A, B} = (A ∪ B))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ∪ cun 2892  {cpr 3351  ∪ cuni 3554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555

This theorem is referenced by:  onun2  4166