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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 7811 | . . . . 5 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | orddisj 6356 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅) |
4 | snnzg 4736 | . . . 4 ⊢ (𝐴 ∈ ω → {𝐴} ≠ ∅) | |
5 | disjpss 4421 | . . . 4 ⊢ (((𝐴 ∩ {𝐴}) = ∅ ∧ {𝐴} ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ {𝐴})) | |
6 | 3, 4, 5 | syl2anc 585 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ (𝐴 ∪ {𝐴})) |
7 | 6 | pssned 4059 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ≠ (𝐴 ∪ {𝐴})) |
8 | df-suc 6324 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | neeq2i 3006 | . 2 ⊢ (𝐴 ≠ suc 𝐴 ↔ 𝐴 ≠ (𝐴 ∪ {𝐴})) |
10 | 7, 9 | sylibr 233 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≠ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∪ cun 3909 ∩ cin 3910 ⊊ wpss 3912 ∅c0 4283 {csn 4587 Ord word 6317 suc csuc 6320 ωcom 7803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-ord 6321 df-on 6322 df-suc 6324 df-om 7804 |
This theorem is referenced by: 1one2o 8593 |
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