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Mirrors > Home > ILE Home > Th. List > minel | GIF version |
Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) |
Ref | Expression |
---|---|
minel | ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inelcm 3343 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅) | |
2 | 1 | necon2bi 2310 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) |
3 | imnan 659 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | sylibr 132 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵)) |
5 | 4 | con2d 589 | . 2 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ 𝐶)) |
6 | 5 | impcom 123 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 ∩ cin 2998 ∅c0 3286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-v 2621 df-dif 3001 df-in 3005 df-nul 3287 |
This theorem is referenced by: unfidisj 6630 hashunlem 10208 |
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