Theorem n0ltsp1le 28589

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 28588	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
2		nnsge1 28567	. . . 4 ⊢ ((𝑁 -s 𝑀) ∈ ℕs → 1s ≤s (𝑁 -s 𝑀))
3		1no 28034	. . . . . . 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 28547	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
7	5, 6	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
8		n0no 28547	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
10	7, 9	subscld 28287	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 -s 𝑀) ∈ No )
11	4, 10, 9	leadds2d 28220	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀))))
12		pncan3s 28297	. . . . . . 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 5123	. . . . 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 28208	. . . . 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 28239	. . . 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 27959	. . 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 2146   class class class wbr 5111  (class class class)co 7416   No csur 27835   <s clts 27836   ≤s cles 27939   1_s c1s 28030   +_s cadds 28183   -_s csubs 28244  ℕ_0scn0s 28536  ℕ_scnns 28537

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27838  df-lts 27839  df-bday 27840  df-les 27940  df-slts 27982  df-cuts 27984  df-0s 28031  df-1s 28032  df-made 28051  df-old 28052  df-left 28054  df-right 28055  df-norec 28162  df-norec2 28173  df-adds 28184  df-negs 28245  df-subs 28246  df-n0s 28538  df-nns 28539

This theorem is used by:  n0lesltp1  28590  bdaypw2n0bndlem  28687  bdaypw2bnd  28689  bdayfinbndlem1  28691