Theorem iocssicc 10363

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 10295	. 2 |- (,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x <_ b ) } )
2		df-icc 10297	. 2 |- [,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x <_ b ) } )
3		xrltle 10200	. 2 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
4		idd 21	. 2 |- ( ( w e. RR* /\ B e. RR* ) -> ( w <_ B -> w <_ B ) )
5	1, 2, 3, 4	ixxssixx 10304	1 |- ( A (,] B ) C_ ( A [,] B )

This proof depends on syntax axioms:   /\ wa 104   e. wcel 2209   C_ wss 3220   class class class wbr 4130  (class class class)co 6085   RR*cxr 8359   < clt 8360   <_ cle 8361   (,]cioc 10291   [,]cicc 10293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-lttrn 8293

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-ioc 10295  df-icc 10297

This theorem is used by: (None)