Theorem topnfbey 30539

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 4279	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		pnfxr 11199	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
3		xrltnr 13070	. . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < +∞
5		zre 12528	. . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ)
6		ltpnf 13071	. . . . . . . 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 13465	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
11	10	fdmi 6680	. . . . . 6 ⊢ dom ... = (ℤ × ℤ)
12	11	ndmov 7551	. . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
13	9, 12	ax-mp 5	. . . 4 ⊢ (0...+∞) = ∅
14	13	eleq2i 2829	. . 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 1542   ∈ wcel 2114  ∅c0 4274  𝒫 cpw 4542   class class class wbr 5086   × cxp 5629  (class class class)co 7367  ℝcr 11037  0cc0 11038  +∞cpnf 11176  ℝ^*cxr 11178   < clt 11179  ℤcz 12524  ...cfz 13461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-neg 11380  df-z 12525  df-fz 13462

This theorem is referenced by: (None)