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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 4344 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 6467 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2828 | . . . . 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 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
7 | eleq1 2827 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V)) | |
8 | 6, 7 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
9 | 8 | con3i 154 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 = ∅) |
10 | sucprc 6462 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
11 | 10 | eqeq1d 2737 | . . . 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 2941 | 1 ⊢ suc 𝐴 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-suc 6392 |
This theorem is referenced by: 0elsuc 7855 peano3 7914 2on0 8521 oelim2 8632 limenpsi 9191 enp1iOLD 9312 ttrclselem2 9764 fseqdom 10064 dfac12lem2 10183 cfsuc 10295 cfpwsdom 10622 rankcf 10815 nosgnn0 27718 sltsolem1 27735 dfrdg2 35777 dfrdg4 35933 dfsucon 43513 ensucne0 43519 ensucne0OLD 43520 |
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