Theorem snunioo 11646

Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
snunioo	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )

Proof of Theorem snunioo

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

Step	Hyp	Ref	Expression
1		simp1 996	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
2		iccid 11574	. . . 4 |- ( A e. RR* -> ( A [,] A ) = { A } )
3	1, 2	syl 16	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A [,] A ) = { A } )
4	3	uneq1d 3657	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( { A } u. ( A (,) B ) ) )
5		simp2 997	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
6		xrleid 11356	. . . 4 |- ( A e. RR* -> A <_ A )
7	1, 6	syl 16	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A )
8		simp3 998	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
9		df-icc 11536	. . . 4 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
10		df-ioo 11533	. . . 4 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		xrltnle 9653	. . . 4 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) )
12		df-ico 11535	. . . 4 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
13		xrlelttr 11359	. . . 4 |- ( ( w e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( w <_ A /\ A < B ) -> w < B ) )
14		xrltle 11355	. . . . . 6 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
15	14	3adant1 1014	. . . . 5 |- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
16	15	adantld 467	. . . 4 |- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( ( A <_ A /\ A < w ) -> A <_ w ) )
17	9, 10, 11, 12, 13, 16	ixxun 11545	. . 3 |- ( ( ( A e. RR* /\ A e. RR* /\ B e. RR* ) /\ ( A <_ A /\ A < B ) ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) )
18	1, 1, 5, 7, 8, 17	syl32anc 1236	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) )
19	4, 18	eqtr3d 2510	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1379   e. wcel 1767   u. cun 3474   {csn 4027   class class class wbr 4447  (class class class)co 6284   RR*cxr 9627   < clt 9628   <_ cle 9629   (,)cioo 11529   [,)cico 11531   [,]cicc 11532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-ioo 11533  df-ico 11535  df-icc 11536

This theorem is referenced by:  prunioo  11649  ioojoin  11651  icombl1  21736  ioombl  21738  tan2h  29652  mbfposadd  29667  itg2addnclem2  29672  ftc1anclem5  29699  iocunico  30811  limciccioolb  31191  cncfiooicclem1  31260  fourierdlem32  31467  fourierdlem93  31528