Theorem unipr 4758

Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)

Hypotheses

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

Assertion

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

Proof of Theorem unipr

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

Step	Hyp	Ref	Expression
1		19.43 1864	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
2		vex 3440	. . . . . . . 8 ⊢ 𝑦 ∈ V
3	2	elpr 4495	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
4	3	anbi2i 622	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
5		andi 1002	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
6	4, 5	bitri 276	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
7	6	exbii 1829	. . . 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 3593	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
10		exancom 1842	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
11	9, 10	bitri 276	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
12		unipr.2	. . . . . . 7 ⊢ 𝐵 ∈ V
13	12	clel3 3593	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
14		exancom 1842	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
15	13, 14	bitri 276	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
16	11, 15	orbi12i 909	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
17	1, 7, 16	3bitr4ri 305	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}))
18	17	abbii 2861	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
19		df-un 3864	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
20		df-uni 4746	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21	18, 19, 20	3eqtr4ri 2830	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 396   ∨ wo 842   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775  Vcvv 3437   ∪ cun 3857  {cpr 4474  ∪ cuni 4745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-un 3864  df-sn 4473  df-pr 4475  df-uni 4746

This theorem is referenced by:  uniprg  4759  uniintsn  4819  uniop  5296  unex  7326  rankxplim  9154  mrcun  16722  indistps  21303  indistps2  21304  leordtval2  21504  ex-uni  27897  mnuprdlem1  40124  mnuprdlem2  40125  mnurndlem1  40133  fouriersw  42078