Theorem elioc1 12217

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

Assertion

Ref	Expression
elioc1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )

Proof of Theorem elioc1

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

Step	Hyp	Ref	Expression
1		df-ioc 12180	. 2 |- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } )
2	1	elixx1 12184	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078  df-ioc 12180

This theorem is referenced by:  ubioc1  12227  elioc2  12236  leordtvallem1  21014  pnfnei  21024  mnfnei  21025  xrge0tsms  22637  lhop1  23777  xrlimcnp  24695  iocinioc2  29541  xrge0tsmsd  29785  xrge0iifcnv  29979  lmxrge0  29998  iocinico  37797  rfcnpre4  39193  iocgtlb  39724  iocleub  39725  eliocd  39730  lenelioc  39763