Theorem elrint 3915

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 3347	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵))
2		elintg 3883	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
3	2	pm5.32i 454	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
4	1, 3	bitri 184	1 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2167  ∀wral 2475   ∩ cin 3156  ∩ cint 3875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-int 3876

This theorem is referenced by:  elrint2  3916