Theorem elrint 3775

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 3223	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵))
2		elintg 3743	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ ∩ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
3	2	pm5.32i 447	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
4	1, 3	bitri 183	1 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1461  ∀wral 2388   ∩ cin 3034  ∩ cint 3735

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-in 3041  df-int 3736

This theorem is referenced by:  elrint2  3776