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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 3282 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅) | |
2 | 1 | necon2bi 2260 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) |
3 | imnan 624 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | sylibr 137 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵)) |
5 | 4 | con2d 554 | . 2 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ 𝐶)) |
6 | 5 | impcom 116 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∩ cin 2916 ∅c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-in 2924 df-nul 3225 |
This theorem is referenced by: (None) |
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