Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3455 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅) | |
2 | 1 | necon2bi 2382 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) |
3 | imnan 680 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵)) | |
4 | 2, 3 | sylibr 133 | . . 3 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵)) |
5 | 4 | con2d 614 | . 2 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ 𝐶)) |
6 | 5 | impcom 124 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 ∩ cin 3101 ∅c0 3395 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-v 2714 df-dif 3104 df-in 3108 df-nul 3396 |
This theorem is referenced by: unfidisj 6868 hashunlem 10689 |
Copyright terms: Public domain | W3C validator |