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Mirrors > Home > ILE Home > Th. List > inelcm | GIF version |
Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.) |
Ref | Expression |
---|---|
inelcm | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3259 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | ne0i 3369 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) | |
3 | 1, 2 | sylbir 134 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ≠ wne 2308 ∩ cin 3070 ∅c0 3363 |
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-in1 603 ax-in2 604 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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-in 3077 df-nul 3364 |
This theorem is referenced by: minel 3424 disjiun 3924 |
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