Theorem n0ltsp1le 28357

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 28356	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
2		nnsge1 28335	. . . 4 ⊢ ((𝑁 -s 𝑀) ∈ ℕs → 1s ≤s (𝑁 -s 𝑀))
3		1no 27802	. . . . . . 7 ⊢ 1s ∈ No
4	3	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 1s ∈ No )
5		simpr 484	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ ℕ0s)
6		n0no 28315	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
8		n0no 28315	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
10	7, 9	subscld 28055	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 -s 𝑀) ∈ No )
11	4, 10, 9	leadds2d 27988	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀))))
12		pncan3s 28065	. . . . . . 7 ⊢ ((𝑀 ∈ No ∧ 𝑁 ∈ No ) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
13	8, 6, 12	syl2an 597	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
14	13	breq2d 5097	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀)) ↔ (𝑀 +s 1s ) ≤s 𝑁))
15	11, 14	bitrd 279	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s 𝑁))
16	2, 15	imbitrid 244	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 -s 𝑀) ∈ ℕs → (𝑀 +s 1s ) ≤s 𝑁))
17	1, 16	sylbid 240	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 → (𝑀 +s 1s ) ≤s 𝑁))
18	8	ad2antrr 727	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 ∈ No )
19		peano2no 27976	. . . . 5 ⊢ (𝑀 ∈ No → (𝑀 +s 1s ) ∈ No )
20	18, 19	syl 17	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → (𝑀 +s 1s ) ∈ No )
21	6	ad2antlr 728	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑁 ∈ No )
22	18	ltsp1d 28007	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 <s (𝑀 +s 1s ))
23		simpr 484	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → (𝑀 +s 1s ) ≤s 𝑁)
24	18, 20, 21, 22, 23	ltlestrd 27728	. . 3 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 <s 𝑁)
25	24	ex 412	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑀 +s 1s ) ≤s 𝑁 → 𝑀 <s 𝑁))
26	17, 25	impbid 212	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  (class class class)co 7367   No csur 27603   <s clts 27604   ≤s cles 27708   1_s c1s 27798   +_s cadds 27951   -_s csubs 28012  ℕ_0scn0s 28304  ℕ_scnns 28305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-n0s 28306  df-nns 28307

This theorem is referenced by:  n0lesltp1  28358  bdaypw2n0bndlem  28455  bdaypw2bnd  28457  bdayfinbndlem1  28459