Theorem renfdisj 8132

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 3509	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2775	. . . . 5 |- x e. _V
3	2	elpr 3654	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 8120	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2431	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 8121	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2431	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 718	. . . 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 2564	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 710   = wceq 1373   e. wcel 2176   i^i cin 3165   (/)c0 3460   {cpr 3634   RRcr 7924   +oocpnf 8104   -oocmnf 8105

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-pnf 8109  df-mnf 8110

This theorem is referenced by: (None)