Theorem unipr 3568

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 1501	. . . 4 ⊢ (∃y((x ∈ y ∧ y = A) ∨ (x ∈ y ∧ y = B)) ↔ (∃y(x ∈ y ∧ y = A) ∨ ∃y(x ∈ y ∧ y = B)))
2		vex 2538	. . . . . . . 8 ⊢ y ∈ V
3	2	elpr 3368	. . . . . . 7 ⊢ (y ∈ {A, B} ↔ (y = A ∨ y = B))
4	3	anbi2i 433	. . . . . 6 ⊢ ((x ∈ y ∧ y ∈ {A, B}) ↔ (x ∈ y ∧ (y = A ∨ y = B)))
5		andi 719	. . . . . 6 ⊢ ((x ∈ y ∧ (y = A ∨ y = B)) ↔ ((x ∈ y ∧ y = A) ∨ (x ∈ y ∧ y = B)))
6	4, 5	bitri 173	. . . . 5 ⊢ ((x ∈ y ∧ y ∈ {A, B}) ↔ ((x ∈ y ∧ y = A) ∨ (x ∈ y ∧ y = B)))
7	6	exbii 1478	. . . 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 2656	. . . . . 6 ⊢ (x ∈ A ↔ ∃y(y = A ∧ x ∈ y))
10		exancom 1481	. . . . . 6 ⊢ (∃y(y = A ∧ x ∈ y) ↔ ∃y(x ∈ y ∧ y = A))
11	9, 10	bitri 173	. . . . 5 ⊢ (x ∈ A ↔ ∃y(x ∈ y ∧ y = A))
12		unipr.2	. . . . . . 7 ⊢ B ∈ V
13	12	clel3 2656	. . . . . 6 ⊢ (x ∈ B ↔ ∃y(y = B ∧ x ∈ y))
14		exancom 1481	. . . . . 6 ⊢ (∃y(y = B ∧ x ∈ y) ↔ ∃y(x ∈ y ∧ y = B))
15	13, 14	bitri 173	. . . . 5 ⊢ (x ∈ B ↔ ∃y(x ∈ y ∧ y = B))
16	11, 15	orbi12i 668	. . . 4 ⊢ ((x ∈ A ∨ x ∈ B) ↔ (∃y(x ∈ y ∧ y = A) ∨ ∃y(x ∈ y ∧ y = B)))
17	1, 7, 16	3bitr4ri 202	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ ∃y(x ∈ y ∧ y ∈ {A, B}))
18	17	abbii 2135	. 2 ⊢ {x ∣ (x ∈ A ∨ x ∈ B)} = {x ∣ ∃y(x ∈ y ∧ y ∈ {A, B})}
19		df-un 2899	. 2 ⊢ (A ∪ B) = {x ∣ (x ∈ A ∨ x ∈ B)}
20		df-uni 3555	. 2 ⊢ ∪ {A, B} = {x ∣ ∃y(x ∈ y ∧ y ∈ {A, B})}
21	18, 19, 20	3eqtr4ri 2053	1 ⊢ ∪ {A, B} = (A ∪ B)

Syntax hints:   ∧ wa 97   ∨ wo 616   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  Vcvv 2535   ∪ cun 2892  {cpr 3351  ∪ cuni 3554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555

This theorem is referenced by:  uniprg  3569  unisn  3570  uniop  3966  unex  4126  bj-unex  7142