Theorem uniprg 2520

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16.

Assertion

Ref	Expression
uniprg	⊢ ((A ∈ C ⋀ B ∈ D) → ∪{A, B} = (A ∪ B))

Proof of Theorem uniprg

Step	Hyp	Ref	Expression
1		preq1 2452	. . . 4 ⊢ (x = A → {x, y} = {A, y})
2	1	unieqd 2516	. . 3 ⊢ (x = A → ∪{x, y} = ∪{A, y})
3		uneq1 2180	. . 3 ⊢ (x = A → (x ∪ y) = (A ∪ y))
4	2, 3	eqeq12d 1492	. 2 ⊢ (x = A → (∪{x, y} = (x ∪ y) ↔ ∪{A, y} = (A ∪ y)))
5		preq2 2453	. . . 4 ⊢ (y = B → {A, y} = {A, B})
6	5	unieqd 2516	. . 3 ⊢ (y = B → ∪{A, y} = ∪{A, B})
7		uneq2 2181	. . 3 ⊢ (y = B → (A ∪ y) = (A ∪ B))
8	6, 7	eqeq12d 1492	. 2 ⊢ (y = B → (∪{A, y} = (A ∪ y) ↔ ∪{A, B} = (A ∪ B)))
9		visset 1816	. . 3 ⊢ x ∈ V
10		visset 1816	. . 3 ⊢ y ∈ V
11	9, 10	unipr 2519	. 2 ⊢ ∪{x, y} = (x ∪ y)
12	4, 8, 11	vtocl2g 1853	1 ⊢ ((A ∈ C ⋀ B ∈ D) → ∪{A, B} = (A ∪ B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960   ∪ cun 2048  {cpr 2414  ∪cuni 2507

This theorem is referenced by:  unctb 7578  unctbOLD 7579  cdrci 10480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-uni 2508

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