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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 3333 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ ∩ 𝐵)) | |
2 | elintg 3870 | . . 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)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ∀wral 2468 ∩ cin 3143 ∩ cint 3862 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-in 3150 df-int 3863 |
This theorem is referenced by: elrint2 3903 |
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