Theorem elicc1 11340

Description: Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
elicc1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ A <_ C /\ C <_ B ) ) )

Proof of Theorem elicc1

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-icc 11303	. 2 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
2	1	elixx1 11305	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,] B ) <-> ( C e. RR* /\ A <_ C /\ C <_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   e. wcel 1761   class class class wbr 4289  (class class class)co 6090   RR*cxr 9413   <_ cle 9415   [,]cicc 11299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-xr 9418  df-icc 11303

This theorem is referenced by:  iccid  11341  iccleub  11347  iccgelb  11348  elicc2  11356  elicc4  11358  elxrge0  11390  lbicc2  11397  ubicc2  11398  difreicc  11413  cnblcld  20313  oprpiece1res1  20482  ovolf  20924  volivth  21046  itg2ge0  21172  itg2const2  21178  taylfvallem1  21781  tayl0  21786  radcnvcl  21841  radcnvle  21844  psercnlem1  21849  eliccelico  26000  xrdifh  26003  xrge0neqmnf  26085  unitssxrge0  26266  esumle  26444  esumlef  26449  esumpinfsum  26462  voliune  26581  volfiniune  26582  ddemeas  26588  prob01  26726  ftc1cnnclem  28390  ftc1anc  28400  ftc2nc  28401  elicc3  28437  iocinico  29512  areaquad  29517