Theorem unisng 1933

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 1816	. . . 4 ⊢ (x = A → {x} = {A})
2	1	unieqd 1929	. . 3 ⊢ (x = A → ∪{x} = ∪{A})
3		id 9	. . 3 ⊢ (x = A → x = A)
4	2, 3	cleq12d 1115	. 2 ⊢ (x = A → (∪{x} = x ↔ ∪{A} = A))
5		visset 1350	. . 3 ⊢ x ∈ V
6	5	unisn 1932	. 2 ⊢ ∪{x} = x
7	4, 6	vtoclg 1383	1 ⊢ (A ∈ B → ∪{A} = A)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  {csn 1808  ∪cuni 1919

This theorem is referenced by:  chsupsn 5313

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-uni 1920

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