Theorem eqv 3215

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Assertion

Ref	Expression
eqv	⊢ (A = V ↔ ∀x x ∈ A)

Distinct variable group:   x,A

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 2016	. 2 ⊢ (A = V ↔ ∀x(x ∈ A ↔ x ∈ V))
2		vex 2536	. . . 4 ⊢ x ∈ V
3	2	tbt 236	. . 3 ⊢ (x ∈ A ↔ (x ∈ A ↔ x ∈ V))
4	3	albii 1339	. 2 ⊢ (∀x x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ V))
5	1, 4	bitr4i 176	1 ⊢ (A = V ↔ ∀x x ∈ A)

Syntax hints:   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2533

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-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2535

This theorem is referenced by:  setindel  4203  dmi  4475