Theorem n0s0suc 28241

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 2733	. . 3 ⊢ (𝑦 = 0s → (𝑦 = 0s ↔ 0s = 0s ))
2		eqeq1 2733	. . . 4 ⊢ (𝑦 = 0s → (𝑦 = (𝑥 +s 1s ) ↔ 0s = (𝑥 +s 1s )))
3	2	rexbidv 3153	. . 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 2733	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 0s ↔ 𝑧 = 0s ))
6		eqeq1 2733	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑥 +s 1s ) ↔ 𝑧 = (𝑥 +s 1s )))
7	6	rexbidv 3153	. . 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 2733	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = 0s ↔ (𝑧 +s 1s ) = 0s ))
10		eqeq1 2733	. . . 4 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = (𝑥 +s 1s ) ↔ (𝑧 +s 1s ) = (𝑥 +s 1s )))
11	10	rexbidv 3153	. . 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 2733	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = 0s ↔ 𝐴 = 0s ))
14		eqeq1 2733	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = (𝑥 +s 1s ) ↔ 𝐴 = (𝑥 +s 1s )))
15	14	rexbidv 3153	. . 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 2729	. . 3 ⊢ 0s = 0s
18	17	orci 865	. 2 ⊢ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))
19		clel5 3620	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
20	19	biimpi 216	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
21		n0sno 28223	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
22		n0sno 28223	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
23		1sno 27742	. . . . . . . 8 ⊢ 1s ∈ No
24		addscan2 27907	. . . . . . . 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 3151	. . . . 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 28232	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  (class class class)co 7349   No csur 27549   0_s c0s 27737   1_s c1s 27738   +_s cadds 27873  ℕ_0scnn0s 28213

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec2 27863  df-adds 27874  df-n0s 28215

This theorem is referenced by:  nnsge1  28242  dfnns2  28268