Theorem uniop 4348

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 3861	. . 3 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
4	3	unieqi 3903	. 2 ⊢ ∪ ⟨𝐴, 𝐵⟩ = ∪ {{𝐴}, {𝐴, 𝐵}}
5	1	snex 4275	. . 3 ⊢ {𝐴} ∈ V
6		prexg 4301	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
7	1, 2, 6	mp2an 426	. . 3 ⊢ {𝐴, 𝐵} ∈ V
8	5, 7	unipr 3907	. 2 ⊢ ∪ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9		snsspr1 3821	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
10		ssequn1 3377	. . 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 2256	1 ⊢ ∪ ⟨𝐴, 𝐵⟩ = {𝐴, 𝐵}

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  {csn 3669  {cpr 3670  ⟨cop 3672  ∪ cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894

This theorem is referenced by:  uniopel  4349  elvvuni  4790  dmrnssfld  4995