Theorem uniop 3966

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 3522	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
4	3	unieqi 3564	. 2 ⊢ ∪ ⟨A, B⟩ = ∪ {{A}, {A, B}}
5		snexgOLD 3909	. . . 4 ⊢ (A ∈ V → {A} ∈ V)
6	1, 5	ax-mp 7	. . 3 ⊢ {A} ∈ V
7		prexgOLD 3920	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
8	1, 2, 7	mp2an 404	. . 3 ⊢ {A, B} ∈ V
9	6, 8	unipr 3568	. 2 ⊢ ∪ {{A}, {A, B}} = ({A} ∪ {A, B})
10		snsspr1 3486	. . 3 ⊢ {A} ⊆ {A, B}
11		ssequn1 3090	. . 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 2046	1 ⊢ ∪ ⟨A, B⟩ = {A, B}

Syntax hints:   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∪ cun 2892   ⊆ wss 2894  {csn 3350  {cpr 3351  ⟨cop 3353  ∪ 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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555

This theorem is referenced by:  uniopel  3967  elvvuni  4331  dmrnssfld  4522