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Mirrors > Home > ILE Home > Th. List > elrint | GIF version |
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
elrint | ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦)) |
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)) |
Colors of variables: wff set class |
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 |
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