Theorem renfdisj 8349

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 3561	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2818	. . . . 5 |- x e. _V
3	2	elpr 3715	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 8337	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2469	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 8338	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2469	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 724	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 121	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 632	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2602	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 716   = wceq 1398   e. wcel 2205   i^i cin 3213   (/)c0 3512   {cpr 3695   RRcr 8142   +oocpnf 8321   -oocmnf 8322

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-pnf 8326  df-mnf 8327

This theorem is referenced by: (None)