Theorem renfdisj 8079

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 3495	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2763	. . . . 5 |- x e. _V
3	2	elpr 3639	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 8067	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2419	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 8068	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2419	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 717	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 121	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 628	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2552	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 709   = wceq 1364   e. wcel 2164   i^i cin 3152   (/)c0 3446   {cpr 3619   RRcr 7871   +oocpnf 8051   -oocmnf 8052

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-pnf 8056  df-mnf 8057

This theorem is referenced by: (None)