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Theorem topnfbey 28025
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 4183 . . 3 ¬ 𝐵 ∈ ∅
2 pnfxr 10494 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 12331 . . . . . . . 8 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 ¬ +∞ < +∞
5 zre 11797 . . . . . . . 8 (+∞ ∈ ℤ → +∞ ∈ ℝ)
6 ltpnf 12332 . . . . . . . 8 (+∞ ∈ ℝ → +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ ℤ → +∞ < +∞)
84, 7mto 189 . . . . . 6 ¬ +∞ ∈ ℤ
98intnan 479 . . . . 5 ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10 fzf 12712 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
1110fdmi 6354 . . . . . 6 dom ... = (ℤ × ℤ)
1211ndmov 7148 . . . . 5 (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
139, 12ax-mp 5 . . . 4 (0...+∞) = ∅
1413eleq2i 2857 . . 3 (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
151, 14mtbir 315 . 2 ¬ 𝐵 ∈ (0...+∞)
1615pm2.21i 117 1 (𝐵 ∈ (0...+∞) → +∞ < 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  c0 4178  𝒫 cpw 4422   class class class wbr 4929   × cxp 5405  (class class class)co 6976  cr 10334  0cc0 10335  +∞cpnf 10471  *cxr 10473   < clt 10474  cz 11793  ...cfz 12708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-pre-lttri 10409  ax-pre-lttrn 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-po 5326  df-so 5327  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-neg 10673  df-z 11794  df-fz 12709
This theorem is referenced by: (None)
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