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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 7630 | . . . . 5 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | orddisj 6229 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅) |
4 | snnzg 4676 | . . . 4 ⊢ (𝐴 ∈ ω → {𝐴} ≠ ∅) | |
5 | disjpss 4361 | . . . 4 ⊢ (((𝐴 ∩ {𝐴}) = ∅ ∧ {𝐴} ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ {𝐴})) | |
6 | 3, 4, 5 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ (𝐴 ∪ {𝐴})) |
7 | 6 | pssned 3999 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ≠ (𝐴 ∪ {𝐴})) |
8 | df-suc 6197 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | neeq2i 2997 | . 2 ⊢ (𝐴 ≠ suc 𝐴 ↔ 𝐴 ≠ (𝐴 ∪ {𝐴})) |
10 | 7, 9 | sylibr 237 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∪ cun 3851 ∩ cin 3852 ⊊ wpss 3854 ∅c0 4223 {csn 4527 Ord word 6190 suc csuc 6193 ωcom 7622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 df-suc 6197 df-om 7623 |
This theorem is referenced by: 1one2o 8349 |
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