Theorem uniprg 3864

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 3709	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	unieqd 3860	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥, 𝑦} = ∪ {𝐴, 𝑦})
3		uneq1 3319	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
4	2, 3	eqeq12d 2219	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) ↔ ∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦)))
5		preq2 3710	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	unieqd 3860	. . 3 ⊢ (𝑦 = 𝐵 → ∪ {𝐴, 𝑦} = ∪ {𝐴, 𝐵})
7		uneq2 3320	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
8	6, 7	eqeq12d 2219	. 2 ⊢ (𝑦 = 𝐵 → (∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦) ↔ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)))
9		vex 2774	. . 3 ⊢ 𝑥 ∈ V
10		vex 2774	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	unipr 3863	. 2 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
12	4, 8, 11	vtocl2g 2836	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175   ∪ cun 3163  {cpr 3633  ∪ cuni 3849

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850

This theorem is referenced by:  onun2  4537  unopn  14448