| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > minel | Structured version Visualization version GIF version | ||
| Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.) |
| Ref | Expression |
|---|---|
| minel | ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inelcm 4465 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅) | |
| 2 | 1 | expcom 413 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 → (𝐶 ∩ 𝐵) ≠ ∅)) |
| 3 | 2 | necon2bd 2956 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐶 ∩ 𝐵) = ∅ → ¬ 𝐴 ∈ 𝐶)) |
| 4 | 3 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-in 3958 df-nul 4334 |
| This theorem is referenced by: peano5 7915 fnsuppres 8216 domunfican 9361 unwdomg 9624 dfac5 10169 ccatval2 14616 mreexexlem2d 17688 hauspwpwf1 23995 |
| Copyright terms: Public domain | W3C validator |