Theorem topnfbey 30501

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

Step	Hyp	Ref	Expression
1		noel 4360	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		pnfxr 11344	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
3		xrltnr 13182	. . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < +∞
5		zre 12643	. . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ)
6		ltpnf 13183	. . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞)
7	5, 6	syl 17	. . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞)
8	4, 7	mto 197	. . . . . 6 ⊢ ¬ +∞ ∈ ℤ
9	8	intnan 486	. . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10		fzf 13571	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
11	10	fdmi 6758	. . . . . 6 ⊢ dom ... = (ℤ × ℤ)
12	11	ndmov 7634	. . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
13	9, 12	ax-mp 5	. . . 4 ⊢ (0...+∞) = ∅
14	13	eleq2i 2836	. . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
15	1, 14	mtbir 323	. 2 ⊢ ¬ 𝐵 ∈ (0...+∞)
16	15	pm2.21i 119	1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∅c0 4352  𝒫 cpw 4622   class class class wbr 5166   × cxp 5698  (class class class)co 7448  ℝcr 11183  0cc0 11184  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324  ℤcz 12639  ...cfz 13567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-pre-lttri 11258  ax-pre-lttrn 11259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-neg 11523  df-z 12640  df-fz 13568

This theorem is referenced by: (None)