Theorem n0ltsp1le 28630

Description: Non-negative surreal ordering relation. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
n0ltsp1le	⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))

Proof of Theorem n0ltsp1le

Step	Hyp	Ref	Expression
1		n0subs2 28629	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
2		nnsge1 28608	. . . 4 ⊢ ((𝑁 -s 𝑀) ∈ ℕs → 1s ≤s (𝑁 -s 𝑀))
3		1no 28075	. . . . . . 7 ⊢ 1s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 1s ∈ No )
5		simpr 490	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ ℕ0s)
6		n0no 28588	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
7	5, 6	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
8		n0no 28588	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
10	7, 9	subscld 28328	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 -s 𝑀) ∈ No )
11	4, 10, 9	leadds2d 28261	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀))))
12		pncan3s 28338	. . . . . . 7 ⊢ ((𝑀 ∈ No ∧ 𝑁 ∈ No ) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
13	8, 6, 12	syl2an 608	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
14	13	breq2d 5115	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀)) ↔ (𝑀 +s 1s ) ≤s 𝑁))
15	11, 14	bitrd 282	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s 𝑁))
16	2, 15	imbitrid 247	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 -s 𝑀) ∈ ℕs → (𝑀 +s 1s ) ≤s 𝑁))
17	1, 16	sylbid 243	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 → (𝑀 +s 1s ) ≤s 𝑁))
18	8	ad2antrr 739	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 ∈ No )
19		peano2no 28249	. . . . 5 ⊢ (𝑀 ∈ No → (𝑀 +s 1s ) ∈ No )
20	18, 19	syl 18	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → (𝑀 +s 1s ) ∈ No )
21	6	ad2antlr 740	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑁 ∈ No )
22	18	ltsp1d 28280	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 <s (𝑀 +s 1s ))
23		simpr 490	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → (𝑀 +s 1s ) ≤s 𝑁)
24	18, 20, 21, 22, 23	ltlestrd 28000	. . 3 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 <s 𝑁)
25	24	ex 418	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑀 +s 1s ) ≤s 𝑁 → 𝑀 <s 𝑁))
26	17, 25	impbid 215	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  (class class class)co 7413   No csur 27876   <s clts 27877   ≤s cles 27980   1_s c1s 28071   +_s cadds 28224   -_s csubs 28285  ℕ_0scn0s 28577  ℕ_scnns 28578

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 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27879  df-lts 27880  df-bday 27881  df-les 27981  df-slts 28023  df-cuts 28025  df-0s 28072  df-1s 28073  df-made 28092  df-old 28093  df-left 28095  df-right 28096  df-norec 28203  df-norec2 28214  df-adds 28225  df-negs 28286  df-subs 28287  df-n0s 28579  df-nns 28580

This theorem is used by:  n0lesltp1  28631  bdaypw2n0bndlem  28728  bdaypw2bnd  28730  bdayfinbndlem1  28732