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

Step	Hyp	Ref	Expression
1		noel 3878	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		pnfxr 9971	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
3		xrltnr 11829	. . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < +∞
5		zre 11258	. . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ)
6		ltpnf 11830	. . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞)
7	5, 6	syl 17	. . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞)
8	4, 7	mto 187	. . . . . 6 ⊢ ¬ +∞ ∈ ℤ
9	8	intnan 951	. . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10		fzf 12201	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
11	10	fdmi 5965	. . . . . 6 ⊢ dom ... = (ℤ × ℤ)
12	11	ndmov 6716	. . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
13	9, 12	ax-mp 5	. . . 4 ⊢ (0...+∞) = ∅
14	13	eleq2i 2680	. . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
15	1, 14	mtbir 312	. 2 ⊢ ¬ 𝐵 ∈ (0...+∞)
16	15	pm2.21i 115	1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵)

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)