Theorem unisn3 4146

Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

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 3428	. . 3 ⊢ (A ∈ B → {x ∈ B ∣ x = A} = {A})
2	1	unieqd 3582	. 2 ⊢ (A ∈ B → ∪ {x ∈ B ∣ x = A} = ∪ {A})
3		unisng 3588	. 2 ⊢ (A ∈ B → ∪ {A} = A)
4	2, 3	eqtrd 2069	1 ⊢ (A ∈ B → ∪ {x ∈ B ∣ x = A} = A)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {crab 2304  {csn 3367  ∪ cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572

This theorem is referenced by: (None)