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Theorem renfdisj 6876
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
StepHypRef Expression
1 disj 3262 . 2  RR  i^i  { +oo , -oo }  (/)  RR  { +oo , -oo }
2 vex 2554 . . . . 5 
_V
32elpr 3385 . . . 4  { +oo , -oo } +oo -oo
4 renepnf 6870 . . . . . 6  RR  =/= +oo
54necon2bi 2254 . . . . 5 +oo  RR
6 renemnf 6871 . . . . . 6  RR  =/= -oo
76necon2bi 2254 . . . . 5 -oo  RR
85, 7jaoi 635 . . . 4 +oo -oo  RR
93, 8sylbi 114 . . 3  { +oo , -oo }  RR
109con2i 557 . 2  RR  { +oo , -oo }
111, 10mprgbir 2373 1  RR 
i^i  { +oo , -oo }  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wo 628   wceq 1242   wcel 1390    i^i cin 2910   (/)c0 3218   {cpr 3368   RRcr 6710   +oocpnf 6854   -oocmnf 6855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-pnf 6859  df-mnf 6860
This theorem is referenced by: (None)
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