Theorem uniprOLD 4928

Description: Obsolete version of unipr 4927 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 1886	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
2		vex 3479	. . . . . . . 8 ⊢ 𝑦 ∈ V
3	2	elpr 4652	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
4	3	anbi2i 624	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
5		andi 1007	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
6	4, 5	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
7	6	exbii 1851	. . . 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 3652	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
10		exancom 1865	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
11	9, 10	bitri 275	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
12		unipr.2	. . . . . . 7 ⊢ 𝐵 ∈ V
13	12	clel3 3652	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
14		exancom 1865	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
15	13, 14	bitri 275	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
16	11, 15	orbi12i 914	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
17	1, 7, 16	3bitr4ri 304	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}))
18	17	abbii 2803	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
19		df-un 3954	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
20		df-uni 4910	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21	18, 19, 20	3eqtr4ri 2772	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 397   ∨ wo 846   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∪ cun 3947  {cpr 4631  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632  df-uni 4910

This theorem is referenced by: (None)