Theorem renfdisj 8019

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 3473	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2742	. . . . 5 |- x e. _V
3	2	elpr 3615	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 8007	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2402	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 8008	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2402	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 716	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 121	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 627	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2535	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 708   = wceq 1353   e. wcel 2148   i^i cin 3130   (/)c0 3424   {cpr 3595   RRcr 7812   +oocpnf 7991   -oocmnf 7992

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-pnf 7996  df-mnf 7997

This theorem is referenced by: (None)