Theorem nnsge1 28235

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 28232	. 2 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ))
2		n0s0suc 28234	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
3		neneq 2931	. . 3 ⊢ (𝑁 ≠ 0s → ¬ 𝑁 = 0s )
4		pm2.53 851	. . . . 5 ⊢ ((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) → (¬ 𝑁 = 0s → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
5	4	imp 406	. . . 4 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ))
6		1sno 27739	. . . . . . . 8 ⊢ 1s ∈ No
7		addslid 27875	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
9		n0sge0 28230	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 0s ≤s 𝑥)
10		n0sno 28216	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
11		0sno 27738	. . . . . . . . . 10 ⊢ 0s ∈ No
12		sleadd1 27896	. . . . . . . . . 10 ⊢ (( 0s ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
13	11, 6, 12	mp3an13 1454	. . . . . . . . 9 ⊢ (𝑥 ∈ No → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
14	10, 13	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
15	9, 14	mpbid 232	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ( 0s +s 1s ) ≤s (𝑥 +s 1s ))
16	8, 15	eqbrtrrid 5143	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s ))
17		breq2 5111	. . . . . 6 ⊢ (𝑁 = (𝑥 +s 1s ) → ( 1s ≤s 𝑁 ↔ 1s ≤s (𝑥 +s 1s )))
18	16, 17	syl5ibrcom 247	. . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁))
19	18	rexlimiv 3127	. . . 4 ⊢ (∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁)
20	5, 19	syl 17	. . 3 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → 1s ≤s 𝑁)
21	2, 3, 20	syl2an 596	. 2 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ) → 1s ≤s 𝑁)
22	1, 21	sylbi 217	1 ⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5107  (class class class)co 7387   No csur 27551   ≤s csle 27656   0_s c0s 27734   1_s c1s 27735   +_s cadds 27866  ℕ_0scnn0s 28206  ℕ_scnns 28207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec2 27856  df-adds 27867  df-n0s 28208  df-nns 28209

This theorem is referenced by:  n0sltp1le  28255