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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 3362 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 4333 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2201 | . . . 4 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
4 | 2, 3 | syl5ibcom 154 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
5 | 1, 4 | mtoi 653 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ suc 𝐴 = ∅) |
6 | 5 | neneqad 2385 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ∅c0 3358 suc csuc 4282 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-dif 3068 df-un 3070 df-nul 3359 df-sn 3528 df-suc 4288 |
This theorem is referenced by: onsucelsucexmid 4440 peano3 4505 frec0g 6287 2on0 6316 zfz1iso 10577 |
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