Theorem reuunisn 2901

Description: A restricted class abstraction with a unique member can be expressed as a singleton.

Assertion

Ref	Expression
reuunisn	⊢ (∃!x ∈ A φ → {x ∈ A∣φ} = {∪{x ∈ A∣φ}})

Proof of Theorem reuunisn

Step	Hyp	Ref	Expression
1		reusn 2898	. 2 ⊢ (∃!x ∈ A φ ↔ ∃y{x ∈ A∣φ} = {y})
2		unieq 2514	. . . . . 6 ⊢ ({x ∈ A∣φ} = {y} → ∪{x ∈ A∣φ} = ∪{y})
3		visset 1816	. . . . . . 7 ⊢ y ∈ V
4	3	unisn 2521	. . . . . 6 ⊢ ∪{y} = y
5	2, 4	syl6eq 1526	. . . . 5 ⊢ ({x ∈ A∣φ} = {y} → ∪{x ∈ A∣φ} = y)
6	5	sneqd 2423	. . . 4 ⊢ ({x ∈ A∣φ} = {y} → {∪{x ∈ A∣φ}} = {y})
7		eqtr3t 1497	. . . 4 ⊢ (({x ∈ A∣φ} = {y} ⋀ {∪{x ∈ A∣φ}} = {y}) → {x ∈ A∣φ} = {∪{x ∈ A∣φ}})
8	6, 7	mpdan 706	. . 3 ⊢ ({x ∈ A∣φ} = {y} → {x ∈ A∣φ} = {∪{x ∈ A∣φ}})
9	8	19.23aiv 1297	. 2 ⊢ (∃y{x ∈ A∣φ} = {y} → {x ∈ A∣φ} = {∪{x ∈ A∣φ}})
10	1, 9	sylbi 199	1 ⊢ (∃!x ∈ A φ → {x ∈ A∣φ} = {∪{x ∈ A∣φ}})

Syntax hints:   → wi 3   = wceq 958  ∃wex 982  ∃!wreu 1650  {crab 1651  {csn 2413  ∪cuni 2507

This theorem is referenced by:  pjspansnt 9495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-clab 1467  df-cleq 1472  df-clel 1475  df-reu 1654  df-rab 1655  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-uni 2508

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