Theorem topnfbey 27325

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 e. ( 0 ... +oo ) -> +oo < B )

Proof of Theorem topnfbey

Step	Hyp	Ref	Expression
1		noel 3919	. . 3 |- -. B e. (/)
2		pnfxr 10092	. . . . . . . 8 |- +oo e. RR*
3		xrltnr 11953	. . . . . . . 8 |- ( +oo e. RR* -> -. +oo < +oo )
4	2, 3	ax-mp 5	. . . . . . 7 |- -. +oo < +oo
5		zre 11381	. . . . . . . 8 |- ( +oo e. ZZ -> +oo e. RR )
6		ltpnf 11954	. . . . . . . 8 |- ( +oo e. RR -> +oo < +oo )
7	5, 6	syl 17	. . . . . . 7 |- ( +oo e. ZZ -> +oo < +oo )
8	4, 7	mto 188	. . . . . 6 |- -. +oo e. ZZ
9	8	intnan 960	. . . . 5 |- -. ( 0 e. ZZ /\ +oo e. ZZ )
10		fzf 12330	. . . . . . 7 |- ... : ( ZZ X. ZZ ) --> ~P ZZ
11	10	fdmi 6052	. . . . . 6 |- dom ... = ( ZZ X. ZZ )
12	11	ndmov 6818	. . . . 5 |- ( -. ( 0 e. ZZ /\ +oo e. ZZ ) -> ( 0 ... +oo ) = (/) )
13	9, 12	ax-mp 5	. . . 4 |- ( 0 ... +oo ) = (/)
14	13	eleq2i 2693	. . 3 |- ( B e. ( 0 ... +oo ) <-> B e. (/) )
15	1, 14	mtbir 313	. 2 |- -. B e. ( 0 ... +oo )
16	15	pm2.21i 116	1 |- ( B e. ( 0 ... +oo ) -> +oo < B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112  (class class class)co 6650   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   ZZcz 11377   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-neg 10269  df-z 11378  df-fz 12327

This theorem is referenced by: (None)