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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 3306 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 4267 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2158 | . . . 4 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
4 | 2, 3 | syl5ibcom 154 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
5 | 1, 4 | mtoi 628 | . 2 ⊢ (𝐴 ∈ 𝑉 → ¬ suc 𝐴 = ∅) |
6 | 5 | neneqad 2341 | 1 ⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 ≠ wne 2262 ∅c0 3302 suc csuc 4216 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-v 2635 df-dif 3015 df-un 3017 df-nul 3303 df-sn 3472 df-suc 4222 |
This theorem is referenced by: onsucelsucexmid 4374 peano3 4439 frec0g 6200 2on0 6229 zfz1iso 10377 |
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