Theorem snunioo 11749

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 1009	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
2		iccid 11671	. . . 4 |- ( A e. RR* -> ( A [,] A ) = { A } )
3	1, 2	syl 17	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A [,] A ) = { A } )
4	3	uneq1d 3555	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( { A } u. ( A (,) B ) ) )
5		simp2 1010	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
6		xrleid 11439	. . . 4 |- ( A e. RR* -> A <_ A )
7	1, 6	syl 17	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A )
8		simp3 1011	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
9		df-icc 11632	. . . 4 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
10		df-ioo 11629	. . . 4 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		xrltnle 9688	. . . 4 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) )
12		df-ico 11631	. . . 4 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
13		xrlelttr 11443	. . . 4 |- ( ( w e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( w <_ A /\ A < B ) -> w < B ) )
14		xrltle 11438	. . . . . 6 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
15	14	3adant1 1027	. . . . 5 |- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
16	15	adantld 473	. . . 4 |- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( ( A <_ A /\ A < w ) -> A <_ w ) )
17	9, 10, 11, 12, 13, 16	ixxun 11641	. . 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 1279	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) )
19	4, 18	eqtr3d 2488	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )

Syntax hints:   -> wi 4   /\ w3a 986   = wceq 1448   e. wcel 1891   u. cun 3370   {csn 3936   class class class wbr 4374  (class class class)co 6276   RR*cxr 9661   < clt 9662   <_ cle 9663   (,)cioo 11625   [,)cico 11627   [,]cicc 11628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-cnex 9582  ax-resscn 9583  ax-pre-lttri 9600  ax-pre-lttrn 9601

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-po 4733  df-so 4734  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-ioo 11629  df-ico 11631  df-icc 11632

This theorem is referenced by:  prunioo  11752  ioojoin  11754  icombl1  22528  ioombl  22530  tan2h  31939  mbfposadd  31990  itg2addnclem2  31996  ftc1anclem5  32023  iocunico  36097  limciccioolb  37743  fourierdlem32  38059  fourierdlem93  38120