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Theorem n0sge0 28341
Description: A non-negative integer is greater than or equal to zero. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
n0sge0 (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Proof of Theorem n0sge0
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5147 . 2 (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2 breq2 5147 . 2 (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3 breq2 5147 . 2 (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4 breq2 5147 . 2 (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5 0sno 27871 . . 3 0s No
6 slerflex 27808 . . 3 ( 0s No → 0s ≤s 0s )
75, 6ax-mp 5 . 2 0s ≤s 0s
85a1i 11 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s No )
9 n0sno 28328 . . . . 5 (𝑚 ∈ ℕ0s𝑚 No )
109adantr 480 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 No )
11 peano2no 28017 . . . . . 6 (𝑚 No → (𝑚 +s 1s ) ∈ No )
129, 11syl 17 . . . . 5 (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ No )
1312adantr 480 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 1s ) ∈ No )
14 simpr 484 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s 𝑚)
159addsridd 27998 . . . . . 6 (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) = 𝑚)
1615adantr 480 . . . . 5 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) = 𝑚)
175a1i 11 . . . . . . . . 9 (⊤ → 0s No )
18 1sno 27872 . . . . . . . . . 10 1s No
1918a1i 11 . . . . . . . . 9 (⊤ → 1s No )
20 0slt1s 27874 . . . . . . . . . 10 0s <s 1s
2120a1i 11 . . . . . . . . 9 (⊤ → 0s <s 1s )
2217, 19, 21sltled 27814 . . . . . . . 8 (⊤ → 0s ≤s 1s )
2322mptru 1547 . . . . . . 7 0s ≤s 1s
245a1i 11 . . . . . . . 8 (𝑚 ∈ ℕ0s → 0s No )
2518a1i 11 . . . . . . . 8 (𝑚 ∈ ℕ0s → 1s No )
2624, 25, 9sleadd2d 28029 . . . . . . 7 (𝑚 ∈ ℕ0s → ( 0s ≤s 1s ↔ (𝑚 +s 0s ) ≤s (𝑚 +s 1s )))
2723, 26mpbii 233 . . . . . 6 (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
2827adantr 480 . . . . 5 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
2916, 28eqbrtrrd 5167 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
308, 10, 13, 14, 29sletrd 27807 . . 3 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s (𝑚 +s 1s ))
3130ex 412 . 2 (𝑚 ∈ ℕ0s → ( 0s ≤s 𝑚 → 0s ≤s (𝑚 +s 1s )))
321, 2, 3, 4, 7, 31n0sind 28337 1 (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wtru 1541  wcel 2108   class class class wbr 5143  (class class class)co 7431   No csur 27684   <s cslt 27685   ≤s csle 27789   0s c0s 27867   1s c1s 27868   +s cadds 27992  0scnn0s 28318
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993  df-n0s 28320
This theorem is referenced by:  nnsgt0  28342  elnns2  28344  nnsge1  28346  n0subs  28360  eln0zs  28386
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