Theorem topnfbey 30729

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 4293	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		pnfxr 11251	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
3		xrltnr 13135	. . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < +∞
5		zre 12586	. . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ)
6		ltpnf 13136	. . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞)
7	5, 6	syl 18	. . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞)
8	4, 7	mto 200	. . . . . 6 ⊢ ¬ +∞ ∈ ℤ
9	8	intnan 491	. . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10		fzf 13530	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
11	10	fdmi 6707	. . . . . 6 ⊢ dom ... = (ℤ × ℤ)
12	11	ndmov 7584	. . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
13	9, 12	ax-mp 5	. . . 4 ⊢ (0...+∞) = ∅
14	13	eleq2i 2857	. . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
15	1, 14	mtbir 326	. 2 ⊢ ¬ 𝐵 ∈ (0...+∞)
16	15	pm2.21i 120	1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105   × cxp 5650  (class class class)co 7400  ℝcr 11087  0cc0 11088  +∞cpnf 11228  ℝ^*cxr 11230   < clt 11231  ℤcz 12582  ...cfz 13526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-pre-lttri 11162  ax-pre-lttrn 11163

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-po 5560  df-so 5561  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-neg 11432  df-z 12583  df-fz 13527

This theorem is referenced by: (None)