Theorem elinti 3936

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elinti	⊢ (A ∈ ∩B → (C ∈ B → A ∈ C))

Proof of Theorem elinti

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 3935	. . 3 ⊢ (A ∈ ∩B → (A ∈ ∩B ↔ ∀x ∈ B A ∈ x))
2		eleq2 2414	. . . 4 ⊢ (x = C → (A ∈ x ↔ A ∈ C))
3	2	rspccv 2953	. . 3 ⊢ (∀x ∈ B A ∈ x → (C ∈ B → A ∈ C))
4	1, 3	syl6bi 219	. 2 ⊢ (A ∈ ∩B → (A ∈ ∩B → (C ∈ B → A ∈ C)))
5	4	pm2.43i 43	1 ⊢ (A ∈ ∩B → (C ∈ B → A ∈ C))

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∩cint 3927

This theorem was proved from 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 theorem 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 referenced by: (None)