Theorem uniprg 4818

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 4629	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	unieqd 4814	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥, 𝑦} = ∪ {𝐴, 𝑦})
3		uneq1 4083	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
4	2, 3	eqeq12d 2814	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦) ↔ ∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦)))
5		preq2 4630	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	unieqd 4814	. . 3 ⊢ (𝑦 = 𝐵 → ∪ {𝐴, 𝑦} = ∪ {𝐴, 𝐵})
7		uneq2 4084	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
8	6, 7	eqeq12d 2814	. 2 ⊢ (𝑦 = 𝐵 → (∪ {𝐴, 𝑦} = (𝐴 ∪ 𝑦) ↔ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)))
9		vex 3444	. . 3 ⊢ 𝑥 ∈ V
10		vex 3444	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	unipr 4817	. 2 ⊢ ∪ {𝑥, 𝑦} = (𝑥 ∪ 𝑦)
12	4, 8, 11	vtocl2g 3520	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  {cpr 4527  ∪ cuni 4800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4801

This theorem is referenced by:  unisng  4819  wunun  10121  tskun  10197  gruun  10217  mrcun  16885  unopn  21508  indistopon  21606  unconn  22034  limcun  24498  sshjval3  29137  prsiga  31500  unelsiga  31503  unelldsys  31527  measxun2  31579  measssd  31584  carsgsigalem  31683  carsgclctun  31689  pmeasmono  31692  probun  31787  indispconn  32594  bj-prmoore  34530  kelac2  40009  mnuund  40986  fourierdlem70  42818  fourierdlem71  42819  saluncl  42959  prsal  42960  meadjun  43101  omeunle  43155