Theorem renfdisj 7824

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 3411	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2689	. . . . 5 |- x e. _V
3	2	elpr 3548	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7813	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2363	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7814	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2363	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 705	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 120	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 616	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2490	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 697   = wceq 1331   e. wcel 1480   i^i cin 3070   (/)c0 3363   {cpr 3528   RRcr 7619   +oocpnf 7797   -oocmnf 7798

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-pnf 7802  df-mnf 7803

This theorem is referenced by: (None)