Theorem uniop 5472

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 4829	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	unieqi 4878	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}}
5		snex 5388	. . 3 ⊢ {𝐴} ∈ V
6		prex 5389	. . 3 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	unipr 4883	. 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
8		snsspr1 4774	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		ssequn1 4140	. . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
10	8, 9	mpbi 229	. 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
11	4, 7, 10	3eqtri 2768	1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ⊆ wss 3910  {csn 4586  {cpr 4588  ⟨cop 4592  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866

This theorem is referenced by:  uniopel  5473  elvvuni  5708  dmrnssfld  5925  dffv2  6936  rankxplim  9814