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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 2941 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) | |
| 2 | nnsuc 7905 | . . . 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 412 | . 2 ⊢ (𝐴 ∈ ω → (¬ 𝐴 = ∅ → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
| 5 | 4 | orrd 864 | 1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∅c0 4333 suc csuc 6386 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 |
| This theorem is referenced by: nnawordex 8675 nnaordex2 8677 nneneq 9246 php 9247 nneneqOLD 9258 phpOLD 9259 cantnfvalf 9705 cantnflt 9712 ttrclselem1 9765 ttrclselem2 9766 hsmexlem9 10465 winainflem 10733 bnj517 34899 nnuni 35727 |
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