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Theorem renfdisj 9683
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 3830 . 2  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<-> 
A. x  e.  RR  -.  x  e.  { +oo , -oo } )
2 renepnf 9677 . . 3  |-  ( x  e.  RR  ->  x  =/= +oo )
3 renemnf 9678 . . 3  |-  ( x  e.  RR  ->  x  =/= -oo )
42, 3nelprd 4015 . 2  |-  ( x  e.  RR  ->  -.  x  e.  { +oo , -oo } )
51, 4mprgbir 2787 1  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1867    i^i cin 3432   (/)c0 3758   {cpr 3995   RRcr 9527   +oocpnf 9661   -oocmnf 9662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667
This theorem is referenced by:  ssxr  9692
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