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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 2746. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
n0i | ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3418 | . . 3 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2234 | . . 3 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅)) | |
3 | 1, 2 | mtbiri 670 | . 2 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
4 | 3 | con2i 622 | 1 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ∈ wcel 2141 ∅c0 3414 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-nul 3415 |
This theorem is referenced by: ne0i 3420 n0ii 3422 unidif0 4151 iin0r 4153 nnm00 6506 dif1enen 6855 enq0tr 7385 |
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