Theorem n0s0suc 28662

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 2764	. . 3 ⊢ (𝑦 = 0s → (𝑦 = 0s ↔ 0s = 0s ))
2		eqeq1 2764	. . . 4 ⊢ (𝑦 = 0s → (𝑦 = (𝑥 +s 1s ) ↔ 0s = (𝑥 +s 1s )))
3	2	rexbidv 3186	. . 3 ⊢ (𝑦 = 0s → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s )))
4	1, 3	orbi12d 932	. 2 ⊢ (𝑦 = 0s → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))))
5		eqeq1 2764	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 0s ↔ 𝑧 = 0s ))
6		eqeq1 2764	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑥 +s 1s ) ↔ 𝑧 = (𝑥 +s 1s )))
7	6	rexbidv 3186	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s )))
8	5, 7	orbi12d 932	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s ))))
9		eqeq1 2764	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = 0s ↔ (𝑧 +s 1s ) = 0s ))
10		eqeq1 2764	. . . 4 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = (𝑥 +s 1s ) ↔ (𝑧 +s 1s ) = (𝑥 +s 1s )))
11	10	rexbidv 3186	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
12	9, 11	orbi12d 932	. 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 2764	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = 0s ↔ 𝐴 = 0s ))
14		eqeq1 2764	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = (𝑥 +s 1s ) ↔ 𝐴 = (𝑥 +s 1s )))
15	14	rexbidv 3186	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
16	13, 15	orbi12d 932	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s ))))
17		eqid 2760	. . 3 ⊢ 0s = 0s
18	17	orci 879	. 2 ⊢ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))
19		clel5 3618	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
20	19	biimpi 219	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
21		n0no 28643	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
22		n0no 28643	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
23		1no 28130	. . . . . . . 8 ⊢ 1s ∈ No
24		addscan2 28313	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
25	23, 24	mp3an3 1479	. . . . . . 7 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
26	21, 22, 25	syl2an 608	. . . . . 6 ⊢ ((𝑧 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
27	26	rexbidva 3184	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥))
28	20, 27	mpbird 260	. . . 4 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s ))
29	28	olcd 888	. . 3 ⊢ (𝑧 ∈ ℕ0s → ((𝑧 +s 1s ) = 0s ∨ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
30	29	a1d 26	. 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 28653	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  (class class class)co 7408   No csur 27931   0_s c0s 28125   1_s c1s 28126   +_s cadds 28279  ℕ_0scn0s 28632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27934  df-lts 27935  df-bday 27936  df-les 28036  df-slts 28078  df-cuts 28080  df-0s 28127  df-1s 28128  df-made 28147  df-old 28148  df-left 28150  df-right 28151  df-norec2 28269  df-adds 28280  df-n0s 28634

This theorem is used by:  nnsge1  28663  dfnns2  28692  bdaypw2n0bnd  28784