Theorem nnsge1 28242

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 28239	. 2 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ))
2		n0s0suc 28241	. . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
3		neneq 2932	. . 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 27746	. . . . . . . 8 ⊢ 1s ∈ No
7		addslid 27882	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
9		n0sge0 28237	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 0s ≤s 𝑥)
10		n0sno 28223	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No )
11		0sno 27745	. . . . . . . . . 10 ⊢ 0s ∈ No
12		sleadd1 27903	. . . . . . . . . 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 5146	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s ))
17		breq2 5114	. . . . . 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 3128	. . . 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 2926  ∃wrex 3054   class class class wbr 5110  (class class class)co 7390   No csur 27558   ≤s csle 27663   0_s c0s 27741   1_s c1s 27742   +_s cadds 27873  ℕ_0scnn0s 28213  ℕ_scnns 28214

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec2 27863  df-adds 27874  df-n0s 28215  df-nns 28216

This theorem is referenced by:  n0sltp1le  28262