Theorem uniop 5534

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 4896	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	unieqi 4943	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}}
5		snex 5451	. . 3 ⊢ {𝐴} ∈ V
6		prex 5452	. . 3 ⊢ {𝐴, 𝐵} ∈ V
7	5, 6	unipr 4948	. 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
8		snsspr1 4839	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
9		ssequn1 4209	. . 3 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
10	8, 9	mpbi 230	. 2 ⊢ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
11	4, 7, 10	3eqtri 2772	1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  {csn 4648  {cpr 4650  ⟨cop 4654  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932

This theorem is referenced by:  uniopel  5535  elvvuni  5776  dmrnssfld  5996  dffv2  7017  rankxplim  9948