Theorem n0s0m1 28509

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.

Step	Hyp	Ref	Expression
1		eqeq1 2769	. . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2		oveq1 7407	. . . 4 ⊢ (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s ))
3	2	eleq1d 2850	. . 3 ⊢ (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 931	. 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)))
5		eqeq1 2769	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s ))
6		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 931	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s)))
9		eqeq1 2769	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s ))
10		oveq1 7407	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2850	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 931	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)))
13		eqeq1 2769	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s ))
14		oveq1 7407	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 931	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)))
17		eqid 2765	. . 3 ⊢ 0s = 0s
18	17	orci 878	. 2 ⊢ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)
19		n0no 28470	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
20		1no 27957	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28219	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 597	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 23	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2865	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)
25	24	olcd 887	. . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
26	25	a1d 26	. 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 28480	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1563   ∈ wcel 2145  (class class class)co 7400   No csur 27758   0_s c0s 27952   1_s c1s 27953   +_s cadds 28106   -_s csubs 28167  ℕ_0scn0s 28459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27761  df-lts 27762  df-bday 27763  df-les 27863  df-slts 27905  df-cuts 27907  df-0s 27954  df-1s 27955  df-made 27974  df-old 27975  df-left 27977  df-right 27978  df-norec 28085  df-norec2 28096  df-adds 28107  df-negs 28168  df-subs 28169  df-n0s 28461

This theorem is referenced by:  n0subs  28510