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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 2944 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
2 | nnsuc 7721 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | |
3 | 1, 2 | sylan2br 595 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) |
4 | 3 | ex 413 | . 2 ⊢ (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
5 | 4 | orrd 860 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∅c0 4257 suc csuc 6262 ωcom 7703 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-tr 5192 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-om 7704 |
This theorem is referenced by: nnawordex 8456 nneneq 8980 php 8981 nneneqOLD 8992 phpOLD 8993 cantnfvalf 9411 cantnflt 9418 ttrclselem1 9471 ttrclselem2 9472 hsmexlem9 10169 winainflem 10437 bnj517 32851 nnuni 33678 |
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