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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 2754. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
ne0i | ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3429 | . 2 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) | |
2 | 1 | neneqad 2426 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ≠ wne 2347 ∅c0 3423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2740 df-dif 3132 df-nul 3424 |
This theorem is referenced by: ne0d 3431 ne0ii 3433 vn0 3434 inelcm 3484 rzal 3521 rexn0 3522 snnzg 3710 prnz 3715 tpnz 3718 brne0 4053 onn0 4401 nn0eln0 4620 ordge1n0im 6437 nnmord 6518 map0g 6688 phpm 6865 fiintim 6928 addclpi 7326 mulclpi 7327 uzn0 9543 iccsupr 9966 ringn0 13237 |
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