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Theorem topnfbey 28257
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.) (Revised 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 4250 . . 3 ¬ 𝐵 ∈ ∅
2 pnfxr 10688 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 12506 . . . . . . . 8 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 ¬ +∞ < +∞
5 zre 11977 . . . . . . . 8 (+∞ ∈ ℤ → +∞ ∈ ℝ)
6 ltpnf 12507 . . . . . . . 8 (+∞ ∈ ℝ → +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ ℤ → +∞ < +∞)
84, 7mto 200 . . . . . 6 ¬ +∞ ∈ ℤ
98intnan 490 . . . . 5 ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10 fzf 12893 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
1110fdmi 6502 . . . . . 6 dom ... = (ℤ × ℤ)
1211ndmov 7316 . . . . 5 (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
139, 12ax-mp 5 . . . 4 (0...+∞) = ∅
1413eleq2i 2884 . . 3 (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
151, 14mtbir 326 . 2 ¬ 𝐵 ∈ (0...+∞)
1615pm2.21i 119 1 (𝐵 ∈ (0...+∞) → +∞ < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  c0 4246  𝒫 cpw 4500   class class class wbr 5033   × cxp 5521  (class class class)co 7139  cr 10529  0cc0 10530  +∞cpnf 10665  *cxr 10667   < clt 10668  cz 11973  ...cfz 12889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-pre-lttri 10604  ax-pre-lttrn 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-neg 10866  df-z 11974  df-fz 12890
This theorem is referenced by: (None)
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