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Mirrors > Home > ILE Home > Th. List > elini | GIF version |
Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
elini.1 | ⊢ 𝐴 ∈ 𝐵 |
elini.2 | ⊢ 𝐴 ∈ 𝐶 |
Ref | Expression |
---|---|
elini | ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elini.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | elini.2 | . 2 ⊢ 𝐴 ∈ 𝐶 | |
3 | elin 3254 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | mpbir2an 926 | 1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ∩ cin 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 |
This theorem is referenced by: exmidonfinlem 7042 taupi 13228 |
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