Theorem topnfbey 31004

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 4283	. . 3 ⊢ ¬ 𝐵 ∈ ∅
2		pnfxr 11335	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
3		xrltnr 13218	. . . . . . . 8 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ¬ +∞ < +∞
5		zre 12667	. . . . . . . 8 ⊢ (+∞ ∈ ℤ → +∞ ∈ ℝ)
6		ltpnf 13219	. . . . . . . 8 ⊢ (+∞ ∈ ℝ → +∞ < +∞)
7	5, 6	syl 18	. . . . . . 7 ⊢ (+∞ ∈ ℤ → +∞ < +∞)
8	4, 7	mto 200	. . . . . 6 ⊢ ¬ +∞ ∈ ℤ
9	8	intnan 492	. . . . 5 ⊢ ¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ)
10		fzf 13613	. . . . . . 7 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ
11	10	fdmi 6709	. . . . . 6 ⊢ dom ... = (ℤ × ℤ)
12	11	ndmov 7593	. . . . 5 ⊢ (¬ (0 ∈ ℤ ∧ +∞ ∈ ℤ) → (0...+∞) = ∅)
13	9, 12	ax-mp 5	. . . 4 ⊢ (0...+∞) = ∅
14	13	eleq2i 2852	. . 3 ⊢ (𝐵 ∈ (0...+∞) ↔ 𝐵 ∈ ∅)
15	1, 14	mtbir 326	. 2 ⊢ ¬ 𝐵 ∈ (0...+∞)
16	15	pm2.21i 120	1 ⊢ (𝐵 ∈ (0...+∞) → +∞ < 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102   × cxp 5645  (class class class)co 7408  ℝcr 11171  0cc0 11172  +∞cpnf 11312  ℝ^*cxr 11314   < clt 11315  ℤcz 12663  ...cfz 13609

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-resscn 11229  ax-pre-lttri 11246  ax-pre-lttrn 11247

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-neg 11516  df-z 12664  df-fz 13610

This theorem is used by: (None)