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| Mirrors > Home > MPE Home > Th. List > nsuceq0 | Structured version Visualization version GIF version | ||
| Description: No successor is empty. (Contributed by NM, 3-Apr-1995.) |
| Ref | Expression |
|---|---|
| nsuceq0 | ⊢ suc 𝐴 ≠ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
| 2 | sucidg 6465 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
| 3 | eleq2 2830 | . . . . 5 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
| 4 | 2, 3 | syl5ibcom 245 | . . . 4 ⊢ (𝐴 ∈ V → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
| 5 | 1, 4 | mtoi 199 | . . 3 ⊢ (𝐴 ∈ V → ¬ suc 𝐴 = ∅) |
| 6 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 7 | eleq1 2829 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V)) | |
| 8 | 6, 7 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
| 9 | 8 | con3i 154 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 = ∅) |
| 10 | sucprc 6460 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
| 11 | 10 | eqeq1d 2739 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (suc 𝐴 = ∅ ↔ 𝐴 = ∅)) |
| 12 | 9, 11 | mtbird 325 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ suc 𝐴 = ∅) |
| 13 | 5, 12 | pm2.61i 182 | . 2 ⊢ ¬ suc 𝐴 = ∅ |
| 14 | 13 | neir 2943 | 1 ⊢ suc 𝐴 ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: 0elsuc 7855 peano3 7913 2on0 8522 oelim2 8633 limenpsi 9192 enp1iOLD 9314 ttrclselem2 9766 fseqdom 10066 dfac12lem2 10185 cfsuc 10297 cfpwsdom 10624 rankcf 10817 nosgnn0 27703 sltsolem1 27720 dfrdg2 35796 dfrdg4 35952 dfsucon 43536 ensucne0 43542 ensucne0OLD 43543 |
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