Theorem uniop 4091

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 3627	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	unieqi 3669	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4026	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4047	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 418	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	unipr 3673	. 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9		snsspr1 3591	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		ssequn1 3171	. . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
11	9, 10	mpbi 144	. 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
12	4, 8, 11	3eqtri 2113	1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}

Syntax hints:   = wceq 1290   ∈ wcel 1439  Vcvv 2620   ∪ cun 2998   ⊆ wss 3000  {csn 3450  {cpr 3451  ⟨cop 3453  ∪ cuni 3659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660

This theorem is referenced by:  uniopel  4092  elvvuni  4515  dmrnssfld  4709