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| Mirrors > Home > MPE Home > Th. List > omsucne | Structured version Visualization version GIF version | ||
| Description: A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.) |
| Ref | Expression |
|---|---|
| omsucne | ⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnord 7895 | . . . . 5 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
| 2 | orddisj 6422 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅) |
| 4 | snnzg 4774 | . . . 4 ⊢ (𝐴 ∈ ω → {𝐴} ≠ ∅) | |
| 5 | disjpss 4461 | . . . 4 ⊢ (((𝐴 ∩ {𝐴}) = ∅ ∧ {𝐴} ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ {𝐴})) | |
| 6 | 3, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ (𝐴 ∪ {𝐴})) |
| 7 | 6 | pssned 4101 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ≠ (𝐴 ∪ {𝐴})) |
| 8 | df-suc 6390 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 9 | 8 | neeq2i 3006 | . 2 ⊢ (𝐴 ≠ suc 𝐴 ↔ 𝐴 ≠ (𝐴 ∪ {𝐴})) |
| 10 | 7, 9 | sylibr 234 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∪ cun 3949 ∩ cin 3950 ⊊ wpss 3952 ∅c0 4333 {csn 4626 Ord word 6383 suc csuc 6386 ωcom 7887 |
| 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-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 df-om 7888 |
| This theorem is referenced by: 1one2o 8684 |
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