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| Mirrors > Home > MPE Home > Th. List > n0s0m1 | Structured version Visualization version GIF version | ||
| Description: Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.) |
| Ref | Expression |
|---|---|
| n0s0m1 | ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq1 2741 | . . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s )) | |
| 2 | oveq1 7438 | . . . 4 ⊢ (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s )) | |
| 3 | 2 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s)) |
| 4 | 1, 3 | orbi12d 919 | . 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s))) |
| 5 | eqeq1 2741 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s )) | |
| 6 | oveq1 7438 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s )) | |
| 7 | 6 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s)) |
| 8 | 5, 7 | orbi12d 919 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s))) |
| 9 | eqeq1 2741 | . . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s )) | |
| 10 | oveq1 7438 | . . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s )) | |
| 11 | 10 | eleq1d 2826 | . . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)) |
| 12 | 9, 11 | orbi12d 919 | . 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))) |
| 13 | eqeq1 2741 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s )) | |
| 14 | oveq1 7438 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s )) | |
| 15 | 14 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s)) |
| 16 | 13, 15 | orbi12d 919 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))) |
| 17 | eqid 2737 | . . 3 ⊢ 0s = 0s | |
| 18 | 17 | orci 866 | . 2 ⊢ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s) |
| 19 | n0sno 28328 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No ) | |
| 20 | 1sno 27872 | . . . . . 6 ⊢ 1s ∈ No | |
| 21 | pncans 28102 | . . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦) | |
| 22 | 19, 20, 21 | sylancl 586 | . . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦) |
| 23 | id 22 | . . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s) | |
| 24 | 22, 23 | eqeltrd 2841 | . . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s) |
| 25 | 24 | olcd 875 | . . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)) |
| 26 | 25 | a1d 25 | . 2 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s) → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))) |
| 27 | 4, 8, 12, 16, 18, 26 | n0sind 28337 | 1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 No csur 27684 0s c0s 27867 1s c1s 27868 +s cadds 27992 -s csubs 28052 ℕ0scnn0s 28318 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-subs 28054 df-n0s 28320 |
| This theorem is referenced by: n0subs 28360 |
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