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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 3255 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 4171 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2142 | . . . 4 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
4 | 2, 3 | syl5ibcom 153 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
5 | 1, 4 | mtoi 622 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ suc 𝐴 = ∅) |
6 | 5 | neneqad 2324 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 ∅c0 3251 suc csuc 4120 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-suc 4126 |
This theorem is referenced by: onsucelsucexmid 4273 peano3 4337 frec0g 6006 2on0 6033 |
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