Theorem iocssicc 8830

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 8762	. 2 |- (,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x <_ b ) } )
2		df-icc 8764	. 2 |- [,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x <_ b ) } )
3		xrltle 8719	. 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 8771	1 |- ( A (,] B ) C_ ( A [,] B )

Syntax hints:   /\ wa 97   e. wcel 1393   C_ wss 2917   class class class wbr 3764  (class class class)co 5512   RR*cxr 7059   < clt 7060   <_ cle 7061   (,]cioc 8758   [,]cicc 8760

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-lttrn 6998

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-ioc 8762  df-icc 8764

This theorem is referenced by: (None)