Theorem uniprg 3854

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 3699	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	unieqd 3850	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥, 𝑦} = ∪ {𝐴, 𝑦})
3		uneq1 3310	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
4	2, 3	eqeq12d 2211	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) ↔ ∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦)))
5		preq2 3700	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	unieqd 3850	. . 3 ⊢ (𝑦 = 𝐵 → ∪ {𝐴, 𝑦} = ∪ {𝐴, 𝐵})
7		uneq2 3311	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
8	6, 7	eqeq12d 2211	. 2 ⊢ (𝑦 = 𝐵 → (∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦) ↔ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)))
9		vex 2766	. . 3 ⊢ 𝑥 ∈ V
10		vex 2766	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	unipr 3853	. 2 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
12	4, 8, 11	vtocl2g 2828	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∪ cun 3155  {cpr 3623  ∪ cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840

This theorem is referenced by:  onun2  4526  unopn  14241