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Theorem nnsge1 28347
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.
StepHypRef Expression
1 elnns 28344 . 2 (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s𝑁 ≠ 0s ))
2 n0s0suc 28346 . . 3 (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )))
3 neneq 2945 . . 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 )))
54imp 406 . . . 4 (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ))
6 1sno 27873 . . . . . . . 8 1s No
7 addslid 28002 . . . . . . . 8 ( 1s No → ( 0s +s 1s ) = 1s )
86, 7ax-mp 5 . . . . . . 7 ( 0s +s 1s ) = 1s
9 n0sge0 28342 . . . . . . . 8 (𝑥 ∈ ℕ0s → 0s ≤s 𝑥)
10 n0sno 28329 . . . . . . . . 9 (𝑥 ∈ ℕ0s𝑥 No )
11 0sno 27872 . . . . . . . . . 10 0s No
12 sleadd1 28023 . . . . . . . . . 10 (( 0s No 𝑥 No ∧ 1s No ) → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
1311, 6, 12mp3an13 1453 . . . . . . . . 9 (𝑥 No → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
1410, 13syl 17 . . . . . . . 8 (𝑥 ∈ ℕ0s → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s )))
159, 14mpbid 232 . . . . . . 7 (𝑥 ∈ ℕ0s → ( 0s +s 1s ) ≤s (𝑥 +s 1s ))
168, 15eqbrtrrid 5178 . . . . . 6 (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s ))
17 breq2 5146 . . . . . 6 (𝑁 = (𝑥 +s 1s ) → ( 1s ≤s 𝑁 ↔ 1s ≤s (𝑥 +s 1s )))
1816, 17syl5ibrcom 247 . . . . 5 (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁))
1918rexlimiv 3147 . . . 4 (∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁)
205, 19syl 17 . . 3 (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → 1s ≤s 𝑁)
212, 3, 20syl2an 596 . 2 ((𝑁 ∈ ℕ0s𝑁 ≠ 0s ) → 1s ≤s 𝑁)
221, 21sylbi 217 1 (𝑁 ∈ ℕs → 1s ≤s 𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1539  wcel 2107  wne 2939  wrex 3069   class class class wbr 5142  (class class class)co 7432   No csur 27685   ≤s csle 27790   0s c0s 27868   1s c1s 27869   +s cadds 27993  0scnn0s 28319  scnns 28320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec2 27983  df-adds 27994  df-n0s 28321  df-nns 28322
This theorem is referenced by: (None)
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