Theorem renfdisj 7979

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 3463	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2733	. . . . 5 |- x e. _V
3	2	elpr 3604	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 7967	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2395	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 7968	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2395	. . . . 5 |- ( x = -oo -> -. x e. RR )
8	5, 7	jaoi 711	. . . 4 |- ( ( x = +oo \/ x = -oo ) -> -. x e. RR )
9	3, 8	sylbi 120	. . 3 |- ( x e. { +oo , -oo } -> -. x e. RR )
10	9	con2i 622	. 2 |- ( x e. RR -> -. x e. { +oo , -oo } )
11	1, 10	mprgbir 2528	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 703   = wceq 1348   e. wcel 2141   i^i cin 3120   (/)c0 3414   {cpr 3584   RRcr 7773   +oocpnf 7951   -oocmnf 7952

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-pnf 7956  df-mnf 7957

This theorem is referenced by: (None)