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Theorem topnfbey 29455
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 4295 . . 3 Β¬ 𝐡 ∈ βˆ…
2 pnfxr 11216 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 13047 . . . . . . . 8 (+∞ ∈ ℝ* β†’ Β¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 Β¬ +∞ < +∞
5 zre 12510 . . . . . . . 8 (+∞ ∈ β„€ β†’ +∞ ∈ ℝ)
6 ltpnf 13048 . . . . . . . 8 (+∞ ∈ ℝ β†’ +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ β„€ β†’ +∞ < +∞)
84, 7mto 196 . . . . . 6 Β¬ +∞ ∈ β„€
98intnan 488 . . . . 5 Β¬ (0 ∈ β„€ ∧ +∞ ∈ β„€)
10 fzf 13435 . . . . . . 7 ...:(β„€ Γ— β„€)βŸΆπ’« β„€
1110fdmi 6685 . . . . . 6 dom ... = (β„€ Γ— β„€)
1211ndmov 7543 . . . . 5 (Β¬ (0 ∈ β„€ ∧ +∞ ∈ β„€) β†’ (0...+∞) = βˆ…)
139, 12ax-mp 5 . . . 4 (0...+∞) = βˆ…
1413eleq2i 2830 . . 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 397   = wceq 1542   ∈ wcel 2107  βˆ…c0 4287  π’« cpw 4565   class class class wbr 5110   Γ— cxp 5636  (class class class)co 7362  β„cr 11057  0cc0 11058  +∞cpnf 11193  β„*cxr 11195   < clt 11196  β„€cz 12506  ...cfz 13431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-pre-lttri 11132  ax-pre-lttrn 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-neg 11395  df-z 12507  df-fz 13432
This theorem is referenced by: (None)
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