Theorem mnfnre 8212

Description: Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
mnfnre	|- -oo e/ RR

Proof of Theorem mnfnre

Step	Hyp	Ref	Expression
1		cnex 8146	. . . . 5 |- CC e. _V
2		2pwuninelg 6444	. . . . 5 |- ( CC e. _V -> -. ~P ~P U. CC e. CC )
3	1, 2	ax-mp 5	. . . 4 |- -. ~P ~P U. CC e. CC
4		df-mnf 8207	. . . . . 6 |- -oo = ~P +oo
5		df-pnf 8206	. . . . . . 7 |- +oo = ~P U. CC
6	5	pweqi 3654	. . . . . 6 |- ~P +oo = ~P ~P U. CC
7	4, 6	eqtri 2250	. . . . 5 |- -oo = ~P ~P U. CC
8	7	eleq1i 2295	. . . 4 |- ( -oo e. CC <-> ~P ~P U. CC e. CC )
9	3, 8	mtbir 675	. . 3 |- -. -oo e. CC
10		recn 8155	. . 3 |- ( -oo e. RR -> -oo e. CC )
11	9, 10	mto 666	. 2 |- -. -oo e. RR
12	11	nelir 2498	1 |- -oo e/ RR

Syntax hints:   -. wn 3   e. wcel 2200   e/ wnel 2495   _Vcvv 2800   ~Pcpw 3650   U.cuni 3891   CCcc 8020   RRcr 8021   +oocpnf 8201   -oocmnf 8202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4633  ax-cnex 8113  ax-resscn 8114

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-ral 2513  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-pnf 8206  df-mnf 8207

This theorem is referenced by:  renemnf  8218  xrltnr  10004  nltmnf  10013