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Mirrors > Home > MPE Home > Th. List > nn0suc | Structured version Visualization version GIF version |
Description: A natural number is either 0 or a successor. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
nn0suc | ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2930 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | nnsuc 7893 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | |
3 | 1, 2 | sylan2br 593 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) |
4 | 3 | ex 411 | . 2 ⊢ (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
5 | 4 | orrd 861 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 ∅c0 4324 suc csuc 6377 ωcom 7875 |
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 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 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 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-om 7876 |
This theorem is referenced by: nnawordex 8666 nnaordex2 8668 nneneq 9246 php 9247 nneneqOLD 9258 phpOLD 9259 cantnfvalf 9704 cantnflt 9711 ttrclselem1 9764 ttrclselem2 9765 hsmexlem9 10464 winainflem 10732 bnj517 34686 nnuni 35497 |
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