Theorem unisng 2522

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.

Assertion

Ref	Expression
unisng	⊢ (A ∈ B → ∪{A} = A)

Proof of Theorem unisng

Step	Hyp	Ref	Expression
1		sneq 2421	. . . 4 ⊢ (x = A → {x} = {A})
2	1	unieqd 2516	. . 3 ⊢ (x = A → ∪{x} = ∪{A})
3		id 59	. . 3 ⊢ (x = A → x = A)
4	2, 3	eqeq12d 1492	. 2 ⊢ (x = A → (∪{x} = x ↔ ∪{A} = A))
5		visset 1816	. . 3 ⊢ x ∈ V
6	5	unisn 2521	. 2 ⊢ ∪{x} = x
7	4, 6	vtoclg 1850	1 ⊢ (A ∈ B → ∪{A} = A)

Syntax hints:   → wi 3   = wceq 958   ∈ wcel 960  {csn 2413  ∪cuni 2507

This theorem is referenced by:  unisn2 2881  unisn3 2882  chsupsn 9307  oefil2 10552

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-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-uni 2508

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