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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 3878 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | sucidg 5720 | . . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
3 | eleq2 2677 | . . . . 5 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅)) | |
4 | 2, 3 | syl5ibcom 234 | . . . 4 ⊢ (𝐴 ∈ V → (suc 𝐴 = ∅ → 𝐴 ∈ ∅)) |
5 | 1, 4 | mtoi 189 | . . 3 ⊢ (𝐴 ∈ V → ¬ suc 𝐴 = ∅) |
6 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
7 | eleq1 2676 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V)) | |
8 | 6, 7 | mpbiri 247 | . . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V) |
9 | 8 | con3i 149 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 = ∅) |
10 | sucprc 5717 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
11 | 10 | eqeq1d 2612 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (suc 𝐴 = ∅ ↔ 𝐴 = ∅)) |
12 | 9, 11 | mtbird 314 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ suc 𝐴 = ∅) |
13 | 5, 12 | pm2.61i 175 | . 2 ⊢ ¬ suc 𝐴 = ∅ |
14 | 13 | neir 2785 | 1 ⊢ suc 𝐴 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-suc 5646 |
This theorem is referenced by: 0elsuc 6927 peano3 6979 2on0 7456 oelim2 7562 limenpsi 8020 enp1i 8080 findcard2 8085 fseqdom 8732 dfac12lem2 8849 cfsuc 8962 cfpwsdom 9285 rankcf 9478 dfrdg2 30945 nosgnn0 31055 sltsolem1 31067 dfrdg4 31228 |
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