Theorem unipr 3585

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 1516	. . . 4 ⊢ (∃y((x ∈ y ∧ y = A) ∨ (x ∈ y ∧ y = B)) ↔ (∃y(x ∈ y ∧ y = A) ∨ ∃y(x ∈ y ∧ y = B)))
2		vex 2554	. . . . . . . 8 ⊢ y ∈ V
3	2	elpr 3385	. . . . . . 7 ⊢ (y ∈ {A, B} ↔ (y = A ∨ y = B))
4	3	anbi2i 430	. . . . . 6 ⊢ ((x ∈ y ∧ y ∈ {A, B}) ↔ (x ∈ y ∧ (y = A ∨ y = B)))
5		andi 730	. . . . . 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 1493	. . . 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 2673	. . . . . 6 ⊢ (x ∈ A ↔ ∃y(y = A ∧ x ∈ y))
10		exancom 1496	. . . . . 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 2673	. . . . . 6 ⊢ (x ∈ B ↔ ∃y(y = B ∧ x ∈ y))
14		exancom 1496	. . . . . 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 680	. . . 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 2150	. 2 ⊢ {x ∣ (x ∈ A ∨ x ∈ B)} = {x ∣ ∃y(x ∈ y ∧ y ∈ {A, B})}
19		df-un 2916	. 2 ⊢ (A ∪ B) = {x ∣ (x ∈ A ∨ x ∈ B)}
20		df-uni 3572	. 2 ⊢ ∪ {A, B} = {x ∣ ∃y(x ∈ y ∧ y ∈ {A, B})}
21	18, 19, 20	3eqtr4ri 2068	1 ⊢ ∪ {A, B} = (A ∪ B)

Syntax hints:   ∧ wa 97   ∨ wo 628   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  Vcvv 2551   ∪ cun 2909  {cpr 3368  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572

This theorem is referenced by:  uniprg  3586  unisn  3587  uniop  3983  unex  4142  bj-unex  9370