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Theorem topnfbey 27213
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 3901 . . 3 ¬ 𝐵 ∈ ∅
2 pnfxr 10052 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 11913 . . . . . . . 8 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 ¬ +∞ < +∞
5 zre 11341 . . . . . . . 8 (+∞ ∈ ℤ → +∞ ∈ ℝ)
6 ltpnf 11914 . . . . . . . 8 (+∞ ∈ ℝ → +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ ℤ → +∞ < +∞)
84, 7mto 188 . . . . . 6 ¬ +∞ ∈ ℤ
98intnan 959 . . . . 5 ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10 fzf 12288 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
1110fdmi 6019 . . . . . 6 dom ... = (ℤ × ℤ)
1211ndmov 6783 . . . . 5 (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
139, 12ax-mp 5 . . . 4 (0...+∞) = ∅
1413eleq2i 2690 . . 3 (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
151, 14mtbir 313 . 2 ¬ 𝐵 ∈ (0...+∞)
1615pm2.21i 116 1 (𝐵 ∈ (0...+∞) → +∞ < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3897  𝒫 cpw 4136   class class class wbr 4623   × cxp 5082  (class class class)co 6615  cr 9895  0cc0 9896  +∞cpnf 10031  *cxr 10033   < clt 10034  cz 11337  ...cfz 12284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-neg 10229  df-z 11338  df-fz 12285
This theorem is referenced by: (None)
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