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Mirrors > Home > ILE Home > Th. List > nsuceq0g | GIF version |
Description: No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.) |
Ref | Expression |
---|---|
nsuceq0g | ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3428 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 4418 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2241 | . . . 4 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
4 | 2, 3 | syl5ibcom 155 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
5 | 1, 4 | mtoi 664 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ suc 𝐴 = ∅) |
6 | 5 | neneqad 2426 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∅c0 3424 suc csuc 4367 |
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 2741 df-dif 3133 df-un 3135 df-nul 3425 df-sn 3600 df-suc 4373 |
This theorem is referenced by: onsucelsucexmid 4531 peano3 4597 frec0g 6400 2on0 6429 zfz1iso 10823 |
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