Theorem n0ltsp1le 28371

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 28370	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
2		nnsge1 28349	. . . 4 ⊢ ((𝑁 -s 𝑀) ∈ ℕs → 1s ≤s (𝑁 -s 𝑀))
3		1no 27816	. . . . . . 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 28329	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
8		n0no 28329	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
10	7, 9	subscld 28069	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 -s 𝑀) ∈ No )
11	4, 10, 9	leadds2d 28002	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀))))
12		pncan3s 28079	. . . . . . 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 5098	. . . . 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 27990	. . . . 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 28021	. . . 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 27742	. . 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 5086  (class class class)co 7360   No csur 27617   <s clts 27618   ≤s cles 27722   1_s c1s 27812   +_s cadds 27965   -_s csubs 28026  ℕ_0scn0s 28318  ℕ_scnns 28319

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-n0s 28320  df-nns 28321

This theorem is referenced by:  n0lesltp1  28372  bdaypw2n0bndlem  28469  bdaypw2bnd  28471  bdayfinbndlem1  28473