Theorem n0s0suc 28363

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 2744	. . 3 ⊢ (𝑦 = 0s → (𝑦 = 0s ↔ 0s = 0s ))
2		eqeq1 2744	. . . 4 ⊢ (𝑦 = 0s → (𝑦 = (𝑥 +s 1s ) ↔ 0s = (𝑥 +s 1s )))
3	2	rexbidv 3185	. . 3 ⊢ (𝑦 = 0s → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s )))
4	1, 3	orbi12d 917	. 2 ⊢ (𝑦 = 0s → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))))
5		eqeq1 2744	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 0s ↔ 𝑧 = 0s ))
6		eqeq1 2744	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑥 +s 1s ) ↔ 𝑧 = (𝑥 +s 1s )))
7	6	rexbidv 3185	. . 3 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s )))
8	5, 7	orbi12d 917	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑧 = (𝑥 +s 1s ))))
9		eqeq1 2744	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = 0s ↔ (𝑧 +s 1s ) = 0s ))
10		eqeq1 2744	. . . 4 ⊢ (𝑦 = (𝑧 +s 1s ) → (𝑦 = (𝑥 +s 1s ) ↔ (𝑧 +s 1s ) = (𝑥 +s 1s )))
11	10	rexbidv 3185	. . 3 ⊢ (𝑦 = (𝑧 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑧 +s 1s ) = (𝑥 +s 1s )))
12	9, 11	orbi12d 917	. 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 2744	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = 0s ↔ 𝐴 = 0s ))
14		eqeq1 2744	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = (𝑥 +s 1s ) ↔ 𝐴 = (𝑥 +s 1s )))
15	14	rexbidv 3185	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
16	13, 15	orbi12d 917	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑦 = (𝑥 +s 1s )) ↔ (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s ))))
17		eqid 2740	. . 3 ⊢ 0s = 0s
18	17	orci 864	. 2 ⊢ ( 0s = 0s ∨ ∃𝑥 ∈ ℕ0s 0s = (𝑥 +s 1s ))
19		clel5 3678	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s ↔ ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
20	19	biimpi 216	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s 𝑧 = 𝑥)
21		n0sno 28346	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
22		n0sno 28346	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
23		1sno 27890	. . . . . . . 8 ⊢ 1s ∈ No
24		addscan2 28044	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
25	23, 24	mp3an3 1450	. . . . . . 7 ⊢ ((𝑧 ∈ No ∧ 𝑥 ∈ No ) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
26	21, 22, 25	syl2an 595	. . . . . 6 ⊢ ((𝑧 ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((𝑧 +s 1s ) = (𝑥 +s 1s ) ↔ 𝑧 = 𝑥))
27	26	rexbidva 3183	. . . . 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 873	. . 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 28355	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  (class class class)co 7448   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011  df-n0s 28338

This theorem is referenced by:  nnsge1  28364  dfnns2  28380