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Theorem topnfbey 27455
Description: Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
topnfbey (𝐵 ∈ (0...+∞) → +∞ < 𝐵)

Proof of Theorem topnfbey
StepHypRef Expression
1 noel 3952 . . 3 ¬ 𝐵 ∈ ∅
2 pnfxr 10130 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 11991 . . . . . . . 8 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 ¬ +∞ < +∞
5 zre 11419 . . . . . . . 8 (+∞ ∈ ℤ → +∞ ∈ ℝ)
6 ltpnf 11992 . . . . . . . 8 (+∞ ∈ ℝ → +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ ℤ → +∞ < +∞)
84, 7mto 188 . . . . . 6 ¬ +∞ ∈ ℤ
98intnan 980 . . . . 5 ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10 fzf 12368 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
1110fdmi 6090 . . . . . 6 dom ... = (ℤ × ℤ)
1211ndmov 6860 . . . . 5 (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
139, 12ax-mp 5 . . . 4 (0...+∞) = ∅
1413eleq2i 2722 . . 3 (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
151, 14mtbir 312 . 2 ¬ 𝐵 ∈ (0...+∞)
1615pm2.21i 116 1 (𝐵 ∈ (0...+∞) → +∞ < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  c0 3948  𝒫 cpw 4191   class class class wbr 4685   × cxp 5141  (class class class)co 6690  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  cz 11415  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-neg 10307  df-z 11416  df-fz 12365
This theorem is referenced by: (None)
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