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Theorem topnfbey 30231
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 (𝐡 ∈ (0...+∞) β†’ +∞ < 𝐡)

Proof of Theorem topnfbey
StepHypRef Expression
1 noel 4325 . . 3 Β¬ 𝐡 ∈ βˆ…
2 pnfxr 11272 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 13105 . . . . . . . 8 (+∞ ∈ ℝ* β†’ Β¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 Β¬ +∞ < +∞
5 zre 12566 . . . . . . . 8 (+∞ ∈ β„€ β†’ +∞ ∈ ℝ)
6 ltpnf 13106 . . . . . . . 8 (+∞ ∈ ℝ β†’ +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ β„€ β†’ +∞ < +∞)
84, 7mto 196 . . . . . 6 Β¬ +∞ ∈ β„€
98intnan 486 . . . . 5 Β¬ (0 ∈ β„€ ∧ +∞ ∈ β„€)
10 fzf 13494 . . . . . . 7 ...:(β„€ Γ— β„€)βŸΆπ’« β„€
1110fdmi 6723 . . . . . 6 dom ... = (β„€ Γ— β„€)
1211ndmov 7588 . . . . 5 (Β¬ (0 ∈ β„€ ∧ +∞ ∈ β„€) β†’ (0...+∞) = βˆ…)
139, 12ax-mp 5 . . . 4 (0...+∞) = βˆ…
1413eleq2i 2819 . . 3 (𝐡 ∈ (0...+∞) ↔ 𝐡 ∈ βˆ…)
151, 14mtbir 323 . 2 Β¬ 𝐡 ∈ (0...+∞)
1615pm2.21i 119 1 (𝐡 ∈ (0...+∞) β†’ +∞ < 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ…c0 4317  π’« cpw 4597   class class class wbr 5141   Γ— cxp 5667  (class class class)co 7405  β„cr 11111  0cc0 11112  +∞cpnf 11249  β„*cxr 11251   < clt 11252  β„€cz 12562  ...cfz 13490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-neg 11451  df-z 12563  df-fz 13491
This theorem is referenced by: (None)
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