Theorem n0s0suc 28511

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 2765	. . 3 ⊢ (𝑦 = 0s → (𝑦 = 0s ↔ 0s = 0s ))
2		eqeq1 2765	. . . 4 ⊢ (𝑦 = 0s → (𝑦 = (𝑥 +s 1s ) ↔ 0s = (𝑥 +s 1s )))
3	2	rexbidv 3187	. . 3 ⊢ (𝑦 = 0s → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s )))
4	1, 3	orbi12d 931	. 2 ⊢ (𝑦 = 0s → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))))
5		eqeq1 2765	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 0s ↔ 𝑧 = 0s ))
6		eqeq1 2765	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑥 +s 1s ) ↔ 𝑧 = (𝑥 +s 1s )))
7	6	rexbidv 3187	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s )))
8	5, 7	orbi12d 931	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s ))))
9		eqeq1 2765	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = 0s ↔ (𝑧 +s 1s ) = 0s ))
10		eqeq1 2765	. . . 4 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = (𝑥 +s 1s ) ↔ (𝑧 +s 1s ) = (𝑥 +s 1s )))
11	10	rexbidv 3187	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
12	9, 11	orbi12d 931	. 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 2765	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = 0s ↔ 𝐴 = 0s ))
14		eqeq1 2765	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = (𝑥 +s 1s ) ↔ 𝐴 = (𝑥 +s 1s )))
15	14	rexbidv 3187	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
16	13, 15	orbi12d 931	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s ))))
17		eqid 2761	. . 3 ⊢ 0s = 0s
18	17	orci 878	. 2 ⊢ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))
19		clel5 3623	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
20	19	biimpi 219	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
21		n0no 28492	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
22		n0no 28492	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
23		1no 27979	. . . . . . . 8 ⊢ 1s ∈ No
24		addscan2 28162	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
25	23, 24	mp3an3 1477	. . . . . . 7 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
26	21, 22, 25	syl2an 607	. . . . . 6 ⊢ ((𝑧 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
27	26	rexbidva 3185	. . . . 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 887	. . 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 28502	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  (class class class)co 7410   No csur 27780   0_s c0s 27974   1_s c1s 27975   +_s cadds 28128  ℕ_0scn0s 28481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-nadd 8651  df-no 27783  df-lts 27784  df-bday 27785  df-les 27885  df-slts 27927  df-cuts 27929  df-0s 27976  df-1s 27977  df-made 27996  df-old 27997  df-left 27999  df-right 28000  df-norec2 28118  df-adds 28129  df-n0s 28483

This theorem is referenced by:  nnsge1  28512  dfnns2  28541  bdaypw2n0bnd  28633