Theorem n0sltp1le 28291

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

Assertion

Ref	Expression
n0sltp1le	⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑀 +s 1s ) ≤s 𝑁))

Proof of Theorem n0sltp1le

Step	Hyp	Ref	Expression
1		n0subs2 28290	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 <s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕs))
2		nnsge1 28271	. . . 4 ⊢ ((𝑁 -s 𝑀) ∈ ℕs → 1s ≤s (𝑁 -s 𝑀))
3		1sno 27771	. . . . . . 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		n0sno 28252	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
8		n0sno 28252	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
10	7, 9	subscld 28003	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑁 -s 𝑀) ∈ No )
11	4, 10, 9	sleadd2d 27939	. . . . 5 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 1s ≤s (𝑁 -s 𝑀) ↔ (𝑀 +s 1s ) ≤s (𝑀 +s (𝑁 -s 𝑀))))
12		pncan3s 28013	. . . . . . 7 ⊢ ((𝑀 ∈ No ∧ 𝑁 ∈ No ) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
13	8, 6, 12	syl2an 596	. . . . . 6 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 +s (𝑁 -s 𝑀)) = 𝑁)
14	13	breq2d 5101	. . . . 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 726	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑀 ∈ No )
19		peano2no 27927	. . . . 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 727	. . . 4 ⊢ (((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) ∧ (𝑀 +s 1s ) ≤s 𝑁) → 𝑁 ∈ No )
22	18	sltp1d 27958	. . . 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	sltletrd 27699	. . 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 1541   ∈ wcel 2111   class class class wbr 5089  (class class class)co 7346   No csur 27578   <s cslt 27579   ≤s csle 27683   1_s c1s 27767   +_s cadds 27902   -_s csubs 27962  ℕ_0scnn0s 28242  ℕ_scnns 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-n0s 28244  df-nns 28245

This theorem is referenced by:  n0sleltp1  28292