Theorem renfdisj 8167

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 3517	. 2 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> A. x e. RR -. x e. { +oo , -oo } )
2		vex 2779	. . . . 5 |- x e. _V
3	2	elpr 3664	. . . 4 |- ( x e. { +oo , -oo } <-> ( x = +oo \/ x = -oo ) )
4		renepnf 8155	. . . . . 6 |- ( x e. RR -> x =/= +oo )
5	4	necon2bi 2433	. . . . 5 |- ( x = +oo -> -. x e. RR )
6		renemnf 8156	. . . . . 6 |- ( x e. RR -> x =/= -oo )
7	6	necon2bi 2433	. . . . 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 2566	1 |- ( RR i^i { +oo , -oo } ) = (/)

Syntax hints:   -. wn 3   \/ wo 710   = wceq 1373   e. wcel 2178   i^i cin 3173   (/)c0 3468   {cpr 3644   RRcr 7959   +oocpnf 8139   -oocmnf 8140

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-pnf 8144  df-mnf 8145

This theorem is referenced by: (None)