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Theorem n0sge0 28356
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 5152 . 2 (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2 breq2 5152 . 2 (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3 breq2 5152 . 2 (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4 breq2 5152 . 2 (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5 0sno 27886 . . 3 0s No
6 slerflex 27823 . . 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 28343 . . . . 5 (𝑚 ∈ ℕ0s𝑚 No )
109adantr 480 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 No )
11 peano2no 28032 . . . . . 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 28013 . . . . . 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 27887 . . . . . . . . . 10 1s No
1918a1i 11 . . . . . . . . 9 (⊤ → 1s No )
20 0slt1s 27889 . . . . . . . . . 10 0s <s 1s
2120a1i 11 . . . . . . . . 9 (⊤ → 0s <s 1s )
2217, 19, 21sltled 27829 . . . . . . . 8 (⊤ → 0s ≤s 1s )
2322mptru 1544 . . . . . . 7 0s ≤s 1s
245a1i 11 . . . . . . . 8 (𝑚 ∈ ℕ0s → 0s No )
2518a1i 11 . . . . . . . 8 (𝑚 ∈ ℕ0s → 1s No )
2624, 25, 9sleadd2d 28044 . . . . . . 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 5172 . . . 4 ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
308, 10, 13, 14, 29sletrd 27822 . . 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 28352 1 (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wtru 1538  wcel 2106   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   ≤s csle 27804   0s c0s 27882   1s c1s 27883   +s cadds 28007  0scnn0s 28333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008  df-n0s 28335
This theorem is referenced by:  nnsgt0  28357  elnns2  28359  nnsge1  28361  n0subs  28375  eln0zs  28401
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