Theorem nnsge1 28735

Description: A positive surreal integer is greater than or equal to one. (Contributed by Scott Fenton, 26-Jul-2025.)

Assertion

Ref	Expression
nnsge1	⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁)

Proof of Theorem nnsge1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnns 28732	. 2 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ))
2		n0s0suc 28734	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
3		neneq 2961	. . 3 ⊢ (𝑁 ≠ 0s → ¬ 𝑁 = 0s )
4		pm2.53 865	. . . . 5 ⊢ ((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) → (¬ 𝑁 = 0s → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
5	4	imp 412	. . . 4 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ))
6		1no 28202	. . . . . . . 8 ⊢ 1s ∈ No
7		addslid 28360	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
9		n0sge0 28730	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 0s ≤s 𝑥)
10		n0no 28715	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No)
11		0no 28201	. . . . . . . . . 10 ⊢ 0s ∈ No
12		leadds1 28381	. . . . . . . . . 10 ⊢ (( 0s ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No) → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
13	11, 6, 12	mp3an13 1481	. . . . . . . . 9 ⊢ (𝑥 ∈ No → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
14	10, 13	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
15	9, 14	mpbid 235	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ( 0s +s 1s ) ≤s (𝑥 +s 1s ))
16	8, 15	eqbrtrrid 5140	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s ))
17		breq2 5106	. . . . . 6 ⊢ (𝑁 = (𝑥 +s 1s ) → ( 1s ≤s 𝑁 ↔ 1s ≤s (𝑥 +s 1s )))
18	16, 17	syl5ibrcom 250	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁))
19	18	rexlimiv 3156	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁)
20	5, 19	syl 18	. . 3 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → 1s ≤s 𝑁)
21	2, 3, 20	syl2an 608	. 2 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ) → 1s ≤s 𝑁)
22	1, 21	sylbi 220	1 ⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   class class class wbr 5102  (class class class)co 7421  Nocsur 28003   ≤s cles 28107   0_s c0s 28197   1_s c1s 28198   +_s cadds 28351  ℕ_0scn0s 28704  ℕ_scnns 28705

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-1s 28200  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec2 28341  df-adds 28352  df-n0s 28706  df-nns 28707

This theorem is used by:  n0ltsp1le  28757