Theorem uniop 4346

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 3859	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	unieqi 3901	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4273	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4299	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	unipr 3905	. 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9		snsspr1 3819	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		ssequn1 3375	. . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
11	9, 10	mpbi 145	. 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
12	4, 8, 11	3eqtri 2254	1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196   ⊆ wss 3198  {csn 3667  {cpr 3668  ⟨cop 3670  ∪ cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892

This theorem is referenced by:  uniopel  4347  elvvuni  4788  dmrnssfld  4993