![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > n0i | GIF version |
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2635. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
n0i | ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3290 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2151 | . . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅)) | |
3 | 1, 2 | mtbiri 635 | . 2 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
4 | 3 | con2i 592 | 1 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1289 ∈ wcel 1438 ∅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-v 2621 df-dif 3001 df-nul 3287 |
This theorem is referenced by: ne0i 3292 unidif0 4002 iin0r 4004 nnm00 6286 dif1enen 6594 enq0tr 6991 |
Copyright terms: Public domain | W3C validator |