Theorem n0s0suc 28291

Description: A non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-Jul-2025.)

Assertion

Ref	Expression
n0s0suc	⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

Distinct variable group:   𝑥,𝐴

Proof of Theorem n0s0suc

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2740	. . 3 ⊢ (𝑦 = 0s → (𝑦 = 0s ↔ 0s = 0s ))
2		eqeq1 2740	. . . 4 ⊢ (𝑦 = 0s → (𝑦 = (𝑥 +s 1s ) ↔ 0s = (𝑥 +s 1s )))
3	2	rexbidv 3165	. . 3 ⊢ (𝑦 = 0s → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s )))
4	1, 3	orbi12d 918	. 2 ⊢ (𝑦 = 0s → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))))
5		eqeq1 2740	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 0s ↔ 𝑧 = 0s ))
6		eqeq1 2740	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑥 +s 1s ) ↔ 𝑧 = (𝑥 +s 1s )))
7	6	rexbidv 3165	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s )))
8	5, 7	orbi12d 918	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s ))))
9		eqeq1 2740	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = 0s ↔ (𝑧 +s 1s ) = 0s ))
10		eqeq1 2740	. . . 4 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = (𝑥 +s 1s ) ↔ (𝑧 +s 1s ) = (𝑥 +s 1s )))
11	10	rexbidv 3165	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
12	9, 11	orbi12d 918	. 2 ⊢ (𝑦 = (𝑧 +s 1s ) → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ ((𝑧 +s 1s ) = 0s ∨ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ))))
13		eqeq1 2740	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = 0s ↔ 𝐴 = 0s ))
14		eqeq1 2740	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = (𝑥 +s 1s ) ↔ 𝐴 = (𝑥 +s 1s )))
15	14	rexbidv 3165	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
16	13, 15	orbi12d 918	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s ))))
17		eqid 2736	. . 3 ⊢ 0s = 0s
18	17	orci 865	. 2 ⊢ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))
19		clel5 3649	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
20	19	biimpi 216	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
21		n0sno 28273	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
22		n0sno 28273	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
23		1sno 27796	. . . . . . . 8 ⊢ 1s ∈ No
24		addscan2 27957	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
25	23, 24	mp3an3 1452	. . . . . . 7 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
26	21, 22, 25	syl2an 596	. . . . . 6 ⊢ ((𝑧 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
27	26	rexbidva 3163	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥))
28	20, 27	mpbird 257	. . . 4 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ))
29	28	olcd 874	. . 3 ⊢ (𝑧 ∈ ℕ0s → ((𝑧 +s 1s ) = 0s ∨ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
30	29	a1d 25	. 2 ⊢ (𝑧 ∈ ℕ0s → ((𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s )) → ((𝑧 +s 1s ) = 0s ∨ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ))))
31	4, 8, 12, 16, 18, 30	n0sind 28282	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  (class class class)co 7410   No csur 27608   0_s c0s 27791   1_s c1s 27792   +_s cadds 27923  ℕ_0scnn0s 28263

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-1s 27794  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec2 27913  df-adds 27924  df-n0s 28265

This theorem is referenced by:  nnsge1  28292  dfnns2  28318