Theorem iocval 10022

Description: Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
iocval	|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) = { x e. RR* | ( A < x /\ x <_ B ) } )

Distinct variable groups:   x, A   x, B

Proof of Theorem iocval

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

Step	Hyp	Ref	Expression
1		df-ioc 9997	. 2 |- (,] = ( y e. RR* , z e. RR* |-> { x e. RR* | ( y < x /\ x <_ z ) } )
2	1	ixxval 10000	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,] B ) = { x e. RR* | ( A < x /\ x <_ B ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {crab 2487   class class class wbr 4043  (class class class)co 5934   RR*cxr 8088   < clt 8089   <_ cle 8090   (,]cioc 9993

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-iota 5229  df-fun 5270  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ioc 9997

This theorem is referenced by:  ioc0  10386