Theorem uniprg 3586

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 3438	. . . 4 ⊢ (x = A → {x, y} = {A, y})
2	1	unieqd 3582	. . 3 ⊢ (x = A → ∪ {x, y} = ∪ {A, y})
3		uneq1 3084	. . 3 ⊢ (x = A → (x ∪ y) = (A ∪ y))
4	2, 3	eqeq12d 2051	. 2 ⊢ (x = A → (∪ {x, y} = (x ∪ y) ↔ ∪ {A, y} = (A ∪ y)))
5		preq2 3439	. . . 4 ⊢ (y = B → {A, y} = {A, B})
6	5	unieqd 3582	. . 3 ⊢ (y = B → ∪ {A, y} = ∪ {A, B})
7		uneq2 3085	. . 3 ⊢ (y = B → (A ∪ y) = (A ∪ B))
8	6, 7	eqeq12d 2051	. 2 ⊢ (y = B → (∪ {A, y} = (A ∪ y) ↔ ∪ {A, B} = (A ∪ B)))
9		vex 2554	. . 3 ⊢ x ∈ V
10		vex 2554	. . 3 ⊢ y ∈ V
11	9, 10	unipr 3585	. 2 ⊢ ∪ {x, y} = (x ∪ y)
12	4, 8, 11	vtocl2g 2611	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∪ {A, B} = (A ∪ B))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ∪ cun 2909  {cpr 3368  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572

This theorem is referenced by:  onun2  4182