Theorem elint2 3934

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
elint2.1	⊢ A ∈ V

Assertion

Ref	Expression
elint2	⊢ (A ∈ ∩B ↔ ∀x ∈ B A ∈ x)

Distinct variable groups:   x,A   x,B

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 ⊢ A ∈ V
2	1	elint 3933	. 2 ⊢ (A ∈ ∩B ↔ ∀x(x ∈ B → A ∈ x))
3		df-ral 2620	. 2 ⊢ (∀x ∈ B A ∈ x ↔ ∀x(x ∈ B → A ∈ x))
4	2, 3	bitr4i 243	1 ⊢ (A ∈ ∩B ↔ ∀x ∈ B A ∈ x)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  ∀wral 2615  Vcvv 2860  ∩cint 3927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-int 3928

This theorem is used by:  elintg  3935  ssint  3943  intssuni  3949  iinuni  4050  dfint3  4319