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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 7884 | . . . . 5 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | orddisj 6412 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ∩ {𝐴}) = ∅) |
4 | snnzg 4783 | . . . 4 ⊢ (𝐴 ∈ ω → {𝐴} ≠ ∅) | |
5 | disjpss 4464 | . . . 4 ⊢ (((𝐴 ∩ {𝐴}) = ∅ ∧ {𝐴} ≠ ∅) → 𝐴 ⊊ (𝐴 ∪ {𝐴})) | |
6 | 3, 4, 5 | syl2anc 582 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ⊊ (𝐴 ∪ {𝐴})) |
7 | 6 | pssned 4098 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ≠ (𝐴 ∪ {𝐴})) |
8 | df-suc 6380 | . . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | neeq2i 3003 | . 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 1533 ∈ wcel 2098 ≠ wne 2937 ∪ cun 3947 ∩ cin 3948 ⊊ wpss 3950 ∅c0 4326 {csn 4632 Ord word 6373 suc csuc 6376 ωcom 7876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377 df-on 6378 df-suc 6380 df-om 7877 |
This theorem is referenced by: 1one2o 8673 |
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