Theorem uniprg 3907

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 3800	. . . 4 ⊢ (x = A → {x, y} = {A, y})
2	1	unieqd 3903	. . 3 ⊢ (x = A → ∪{x, y} = ∪{A, y})
3		uneq1 3412	. . 3 ⊢ (x = A → (x ∪ y) = (A ∪ y))
4	2, 3	eqeq12d 2367	. 2 ⊢ (x = A → (∪{x, y} = (x ∪ y) ↔ ∪{A, y} = (A ∪ y)))
5		preq2 3801	. . . 4 ⊢ (y = B → {A, y} = {A, B})
6	5	unieqd 3903	. . 3 ⊢ (y = B → ∪{A, y} = ∪{A, B})
7		uneq2 3413	. . 3 ⊢ (y = B → (A ∪ y) = (A ∪ B))
8	6, 7	eqeq12d 2367	. 2 ⊢ (y = B → (∪{A, y} = (A ∪ y) ↔ ∪{A, B} = (A ∪ B)))
9		vex 2863	. . 3 ⊢ x ∈ V
10		vex 2863	. . 3 ⊢ y ∈ V
11	9, 10	unipr 3906	. 2 ⊢ ∪{x, y} = (x ∪ y)
12	4, 8, 11	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → ∪{A, B} = (A ∪ B))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3208  {cpr 3739  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893

This theorem is referenced by: (None)