Theorem uniop 3983

Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

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

Assertion

Ref	Expression
uniop	⊢ ∪ ⟨A, B⟩ = {A, B}

Proof of Theorem uniop

Step	Hyp	Ref	Expression
1		opthw.1	. . . 4 ⊢ A ∈ V
2		opthw.2	. . . 4 ⊢ B ∈ V
3	1, 2	dfop 3539	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
4	3	unieqi 3581	. 2 ⊢ ∪ ⟨A, B⟩ = ∪ {{A}, {A, B}}
5		snexgOLD 3926	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
6	1, 5	ax-mp 7	. . 3 ⊢ {A} ∈ V
7		prexgOLD 3937	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
8	1, 2, 7	mp2an 402	. . 3 ⊢ {A, B} ∈ V
9	6, 8	unipr 3585	. 2 ⊢ ∪ {{A}, {A, B}} = ({A} ∪ {A, B})
10		snsspr1 3503	. . 3 ⊢ {A} ⊆ {A, B}
11		ssequn1 3107	. . 3 ⊢ ({A} ⊆ {A, B} ↔ ({A} ∪ {A, B}) = {A, B})
12	10, 11	mpbi 133	. 2 ⊢ ({A} ∪ {A, B}) = {A, B}
13	4, 9, 12	3eqtri 2061	1 ⊢ ∪ ⟨A, B⟩ = {A, B}

Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {csn 3367  {cpr 3368  ⟨cop 3370  ∪ 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572

This theorem is referenced by:  uniopel  3984  elvvuni  4347  dmrnssfld  4538