Theorem unisn3 2876

Description: Union of a singleton in the form of a restricted class abstraction.

Assertion

Ref	Expression
unisn3	⊢ (A ∈ B → ∪{x ∈ B∣x = A} = A)

Distinct variable groups:   x,A   x,B

Proof of Theorem unisn3

Step	Hyp	Ref	Expression
1		rabsn 2445	. . 3 ⊢ (A ∈ B → {x ∈ B∣x = A} = {A})
2	1	unieqd 2512	. 2 ⊢ (A ∈ B → ∪{x ∈ B∣x = A} = ∪{A})
3		unisng 2518	. 2 ⊢ (A ∈ B → ∪{A} = A)
4	2, 3	eqtrd 1507	1 ⊢ (A ∈ B → ∪{x ∈ B∣x = A} = A)

Syntax hints:   → wi 3   = wceq 956   ∈ wcel 958  {crab 1648  {csn 2409  ∪cuni 2503

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-uni 2504

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