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Theorem xnn0lenn0nn0 9616
Description: An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)
Assertion
Ref Expression
xnn0lenn0nn0 ((𝑀 ∈ ℕ0*𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℕ0)

Proof of Theorem xnn0lenn0nn0
StepHypRef Expression
1 elxnn0 9010 . . 3 (𝑀 ∈ ℕ0* ↔ (𝑀 ∈ ℕ0𝑀 = +∞))
2 2a1 25 . . . 4 (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝑀𝑁𝑀 ∈ ℕ0)))
3 breq1 3902 . . . . . . 7 (𝑀 = +∞ → (𝑀𝑁 ↔ +∞ ≤ 𝑁))
43adantr 274 . . . . . 6 ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ +∞ ≤ 𝑁))
5 nn0re 8954 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
65rexrd 7783 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℝ*)
7 xgepnf 9567 . . . . . . . . 9 (𝑁 ∈ ℝ* → (+∞ ≤ 𝑁𝑁 = +∞))
86, 7syl 14 . . . . . . . 8 (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁𝑁 = +∞))
9 pnfnre 7775 . . . . . . . . 9 +∞ ∉ ℝ
10 eleq1 2180 . . . . . . . . . . 11 (𝑁 = +∞ → (𝑁 ∈ ℕ0 ↔ +∞ ∈ ℕ0))
11 nn0re 8954 . . . . . . . . . . . 12 (+∞ ∈ ℕ0 → +∞ ∈ ℝ)
12 elnelall 2392 . . . . . . . . . . . 12 (+∞ ∈ ℝ → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0))
1311, 12syl 14 . . . . . . . . . . 11 (+∞ ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0))
1410, 13syl6bi 162 . . . . . . . . . 10 (𝑁 = +∞ → (𝑁 ∈ ℕ0 → (+∞ ∉ ℝ → 𝑀 ∈ ℕ0)))
1514com13 80 . . . . . . . . 9 (+∞ ∉ ℝ → (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0)))
169, 15ax-mp 5 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 = +∞ → 𝑀 ∈ ℕ0))
178, 16sylbid 149 . . . . . . 7 (𝑁 ∈ ℕ0 → (+∞ ≤ 𝑁𝑀 ∈ ℕ0))
1817adantl 275 . . . . . 6 ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (+∞ ≤ 𝑁𝑀 ∈ ℕ0))
194, 18sylbid 149 . . . . 5 ((𝑀 = +∞ ∧ 𝑁 ∈ ℕ0) → (𝑀𝑁𝑀 ∈ ℕ0))
2019ex 114 . . . 4 (𝑀 = +∞ → (𝑁 ∈ ℕ0 → (𝑀𝑁𝑀 ∈ ℕ0)))
212, 20jaoi 690 . . 3 ((𝑀 ∈ ℕ0𝑀 = +∞) → (𝑁 ∈ ℕ0 → (𝑀𝑁𝑀 ∈ ℕ0)))
221, 21sylbi 120 . 2 (𝑀 ∈ ℕ0* → (𝑁 ∈ ℕ0 → (𝑀𝑁𝑀 ∈ ℕ0)))
23223imp 1160 1 ((𝑀 ∈ ℕ0*𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℕ0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 682  w3a 947   = wceq 1316  wcel 1465  wnel 2380   class class class wbr 3899  cr 7587  +∞cpnf 7765  *cxr 7767  cle 7769  0cn0 8945  0*cxnn0 9008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-rnegex 7697  ax-pre-ltirr 7700
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-inn 8689  df-n0 8946  df-xnn0 9009
This theorem is referenced by:  xnn0le2is012  9617
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