Theorem renfdisj 7239

Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Assertion

Ref	Expression
renfdisj	|- ( RR i^i { +oo , -oo } ) = (/)

Proof of Theorem renfdisj

Step	Hyp	Ref	Expression
1		disj 3299	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2605	. . . . 5 |- x e. _V
3	2	elpr 3427	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7228	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2301	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7229	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2301	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 669	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 119	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 590	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2422	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 662   = wceq 1285   e. wcel 1434   i^i cin 2973   (/)c0 3258   {cpr 3407   RRcr 7042   +oocpnf 7212   -oocmnf 7213

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-pnf 7217  df-mnf 7218

This theorem is referenced by: (None)