Theorem renfdisj 7172

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 3292	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2604	. . . . 5 |- x e. _V
3	2	elpr 3419	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7166	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2300	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7167	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2300	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 668	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 119	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 589	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2421	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 661   = wceq 1284   e. wcel 1433   i^i cin 2972   (/)c0 3251   {cpr 3399   RRcr 6980   +oocpnf 7150   -oocmnf 7151

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-pnf 7155  df-mnf 7156

This theorem is referenced by: (None)