Theorem iocssicc 12261

Description: A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)

Assertion

Ref	Expression
iocssicc	|- ( A (,] B ) C_ ( A [,] B )

Proof of Theorem iocssicc

Dummy variables a b w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioc 12180	. 2 |- (,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x <_ b ) } )
2		df-icc 12182	. 2 |- [,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x <_ b ) } )
3		xrltle 11982	. 2 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
4		idd 24	. 2 |- ( ( w e. RR* /\ B e. RR* ) -> ( w <_ B -> w <_ B ) )
5	1, 2, 3, 4	ixxssixx 12189	1 |- ( A (,] B ) C_ ( A [,] B )

Syntax hints:   /\ wa 384   e. wcel 1990   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176   [,]cicc 12178

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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioc 12180  df-icc 12182

This theorem is referenced by:  xrge0iifcnv  29979  xrge0iifcv  29980  xrge0iifhom  29983  pnfneige0  29997  lmxrge0  29998  eliccelioc  39747  limcicciooub  39869  fourierdlem17  40341  fourierdlem35  40359  fourierdlem41  40365  fourierdlem48  40371  fourierdlem49  40372  fourierdlem51  40374  fourierdlem71  40394  fourierdlem102  40425  fourierdlem114  40437