Theorem renfdisj 7848

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 3416	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2692	. . . . 5 |- x e. _V
3	2	elpr 3553	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7837	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2364	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7838	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2364	. . . . 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 2493	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 698   = wceq 1332   e. wcel 1481   i^i cin 3075   (/)c0 3368   {cpr 3533   RRcr 7643   +oocpnf 7821   -oocmnf 7822

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-pnf 7826  df-mnf 7827

This theorem is referenced by: (None)