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Theorem n0s0m1 28359
Description: Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)
Assertion
Ref Expression
n0s0m1 (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))

Proof of Theorem n0s0m1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2741 . . 3 (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2 oveq1 7438 . . . 4 (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s ))
32eleq1d 2826 . . 3 (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s))
41, 3orbi12d 919 . 2 (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)))
5 eqeq1 2741 . . 3 (𝑥 = 𝑦 → (𝑥 = 0s𝑦 = 0s ))
6 oveq1 7438 . . . 4 (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
76eleq1d 2826 . . 3 (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s))
85, 7orbi12d 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 ))
1110eleq1d 2826 . . 3 (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
129, 11orbi12d 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 = 0s𝐴 = 0s ))
14 oveq1 7438 . . . 4 (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
1514eleq1d 2826 . . 3 (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s))
1613, 15orbi12d 919 . 2 (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)))
17 eqid 2737 . . 3 0s = 0s
1817orci 866 . 2 ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)
19 n0sno 28328 . . . . . 6 (𝑦 ∈ ℕ0s𝑦 No )
20 1sno 27872 . . . . . 6 1s No
21 pncans 28102 . . . . . 6 ((𝑦 No ∧ 1s No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
2219, 20, 21sylancl 586 . . . . 5 (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23 id 22 . . . . 5 (𝑦 ∈ ℕ0s𝑦 ∈ ℕ0s)
2422, 23eqeltrd 2841 . . . 4 (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)
2524olcd 875 . . 3 (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
2625a1d 25 . 2 (𝑦 ∈ ℕ0s → ((𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s) → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)))
274, 8, 12, 16, 18, 26n0sind 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  0scnn0s 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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