Theorem unipr 3810

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 1621	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
2		vex 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
3	2	elpr 3604	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
4	3	anbi2i 454	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)))
5		andi 813	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
6	4, 5	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
7	6	exbii 1598	. . . 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 2865	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
10		exancom 1601	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
11	9, 10	bitri 183	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴))
12		unipr.2	. . . . . . 7 ⊢ 𝐵 ∈ V
13	12	clel3 2865	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
14		exancom 1601	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
15	13, 14	bitri 183	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵))
16	11, 15	orbi12i 759	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = 𝐵)))
17	1, 7, 16	3bitr4ri 212	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵}))
18	17	abbii 2286	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
19		df-un 3125	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
20		df-uni 3797	. 2 ⊢ ∪ {𝐴, 𝐵} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ {𝐴, 𝐵})}
21	18, 19, 20	3eqtr4ri 2202	1 ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 103   ∨ wo 703   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  Vcvv 2730   ∪ cun 3119  {cpr 3584  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  uniprg  3811  unisn  3812  uniop  4240  unex  4426  bj-unex  13954