Theorem nnsge1 28711

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 28708	. 2 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ))
2		n0s0suc 28710	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
3		neneq 2962	. . 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 28178	. . . . . . . 8 ⊢ 1s ∈ No
7		addslid 28336	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
9		n0sge0 28706	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 0s ≤s 𝑥)
10		n0no 28691	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
11		0no 28177	. . . . . . . . . 10 ⊢ 0s ∈ No
12		leadds1 28357	. . . . . . . . . 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 5141	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s ))
17		breq2 5107	. . . . . 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 3157	. . . 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 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   class class class wbr 5103  (class class class)co 7412   No csur 27979   ≤s cles 28083   0_s c0s 28173   1_s c1s 28174   +_s cadds 28327  ℕ_0scn0s 28680  ℕ_scnns 28681

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-1s 28176  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec2 28317  df-adds 28328  df-n0s 28682  df-nns 28683

This theorem is used by:  n0ltsp1le  28733