Theorem elrint 4956

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
elrint	⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Distinct variable groups:   𝑦,𝐵   𝑦,𝑋

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elrint

Step	Hyp	Ref	Expression
1		elin 3933	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵))
2		elintg 4921	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
3	2	pm5.32i 574	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
4	1, 3	bitri 275	1 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3045   ∩ cin 3916  ∩ cint 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-in 3924  df-int 4914

This theorem is referenced by:  elrint2  4957  ptcnplem  23515  tmdgsum2  23990  limciun  25802