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Theorem topnfbey 26717
 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 3878 . . 3 ¬ 𝐵 ∈ ∅
2 pnfxr 9971 . . . . . . . 8 +∞ ∈ ℝ*
3 xrltnr 11829 . . . . . . . 8 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
42, 3ax-mp 5 . . . . . . 7 ¬ +∞ < +∞
5 zre 11258 . . . . . . . 8 (+∞ ∈ ℤ → +∞ ∈ ℝ)
6 ltpnf 11830 . . . . . . . 8 (+∞ ∈ ℝ → +∞ < +∞)
75, 6syl 17 . . . . . . 7 (+∞ ∈ ℤ → +∞ < +∞)
84, 7mto 187 . . . . . 6 ¬ +∞ ∈ ℤ
98intnan 951 . . . . 5 ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10 fzf 12201 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
1110fdmi 5965 . . . . . 6 dom ... = (ℤ × ℤ)
1211ndmov 6716 . . . . 5 (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
139, 12ax-mp 5 . . . 4 (0...+∞) = ∅
1413eleq2i 2680 . . 3 (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
151, 14mtbir 312 . 2 ¬ 𝐵 ∈ (0...+∞)
1615pm2.21i 115 1 (𝐵 ∈ (0...+∞) → +∞ < 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  (class class class)co 6549  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953  ℤcz 11254  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-neg 10148  df-z 11255  df-fz 12198 This theorem is referenced by: (None)
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