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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 3342 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | ne0i 3453 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) | |
3 | 1, 2 | sylbir 135 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2164 ≠ wne 2364 ∩ cin 3152 ∅c0 3446 |
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-in1 615 ax-in2 616 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 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-v 2762 df-dif 3155 df-in 3159 df-nul 3447 |
This theorem is referenced by: minel 3508 disjiun 4024 |
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