Theorem renfdisj 7958

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 3457	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2729	. . . . 5 |- x e. _V
3	2	elpr 3597	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7946	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2391	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7947	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2391	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 706	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 120	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 617	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2524	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 698   = wceq 1343   e. wcel 2136   i^i cin 3115   (/)c0 3409   {cpr 3577   RRcr 7752   +oocpnf 7930   -oocmnf 7931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-pnf 7935  df-mnf 7936

This theorem is referenced by: (None)