Theorem class2seteq 2730

Description: Equality theorem based on class2set 2729. (The proof was shortened by Raph Levien, 30-Jun-2006.)

Assertion

Ref	Expression
class2seteq	⊢ (A ∈ B → {x ∈ A∣A ∈ V} = A)

Distinct variable group:   x,A

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elisset 1813	. 2 ⊢ (A ∈ B → A ∈ V)
2		ax-1 4	. . . . 5 ⊢ (A ∈ V → (x ∈ A → A ∈ V))
3	2	r19.21aiv 1710	. . . 4 ⊢ (A ∈ V → ∀x ∈ A A ∈ V)
4		rabid2 1767	. . . 4 ⊢ (A = {x ∈ A∣A ∈ V} ↔ ∀x ∈ A A ∈ V)
5	3, 4	sylibr 200	. . 3 ⊢ (A ∈ V → A = {x ∈ A∣A ∈ V})
6	5	eqcomd 1477	. 2 ⊢ (A ∈ V → {x ∈ A∣A ∈ V} = A)
7	1, 6	syl 10	1 ⊢ (A ∈ B → {x ∈ A∣A ∈ V} = A)

Syntax hints:   → wi 3   = wceq 954   ∈ wcel 956  ∀wral 1642  {crab 1645  Vcvv 1807

This theorem is referenced by:  fsum1s 6955  fsump1s 6959

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rab 1649  df-v 1808

Copyright terms: Public domain