Theorem uniprOLD 4869

Description: Obsolete version of unipr 4868 as of 1-Sep-2024. (Contributed by NM, 23-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
unipr.1	⊢ 𝐴 ∈ V
unipr.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
uniprOLD	⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Proof of Theorem uniprOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.43 1884	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
2		vex 3445	. . . . . . . 8 ⊢ 𝑦 ∈ V
3	2	elpr 4594	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
4	3	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
5		andi 1005	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
6	4, 5	bitri 274	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
7	6	exbii 1849	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
8		unipr.1	. . . . . . 7 ⊢ 𝐴 ∈ V
9	8	clel3 3602	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
10		exancom 1863	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
11	9, 10	bitri 274	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
12		unipr.2	. . . . . . 7 ⊢ 𝐵 ∈ V
13	12	clel3 3602	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
14		exancom 1863	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
15	13, 14	bitri 274	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
16	11, 15	orbi12i 912	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
17	1, 7, 16	3bitr4ri 303	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}))
18	17	abbii 2807	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
19		df-un 3902	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
20		df-uni 4851	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21	18, 19, 20	3eqtr4ri 2776	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 396   ∨ wo 844   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2714  Vcvv 3441   ∪ cun 3895  {cpr 4573  ∪ cuni 4850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-un 3902  df-sn 4572  df-pr 4574  df-uni 4851

This theorem is referenced by: (None)