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Mirrors > Home > ILE Home > Th. List > ne0i | GIF version |
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2702. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
ne0i | ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3368 | . 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) | |
2 | 1 | neneqad 2387 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ≠ wne 2308 ∅c0 3363 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-nul 3364 |
This theorem is referenced by: ne0d 3370 ne0ii 3372 vn0 3373 inelcm 3423 rzal 3460 rexn0 3461 snnzg 3640 prnz 3645 tpnz 3648 brne0 3977 onn0 4322 nn0eln0 4533 ordge1n0im 6333 nnmord 6413 map0g 6582 phpm 6759 fiintim 6817 addclpi 7135 mulclpi 7136 uzn0 9341 iccsupr 9749 |
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