Theorem elinti 4916

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 4915	. . 3 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥))
2		eleq2 2850	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐶))
3	2	rspccv 3574	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))
4	1, 3	biimtrdi 256	. 2 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)))
5	4	pm2.43i 53	1 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3077  ∩ cint 4907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-int 4908

This theorem is used by:  inttsk  10859  subgint  19361  subrngint  20812  subrgint  20847  lssintcl  21239  ufinffr  24248  shintcli  31931  intlidl  33970  insiga  34770  intsal  47339