Theorem elinti 4959

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 4958	. . 3 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
2		eleq2 2821	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐶))
3	2	rspccv 3609	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))
4	1, 3	biimtrdi 252	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)))
5	4	pm2.43i 52	1 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∈ wcel 2105  ∀wral 3060  ∩ cint 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-int 4951

This theorem is referenced by:  inttsk  10775  subgint  19073  subrngint  20456  subrgint  20493  lssintcl  20807  ufinffr  23754  shintcli  31017  intlidl  32978  insiga  33601  intsal  45508