Theorem snunioo2 39731

Description: The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
snunioo2	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )

Proof of Theorem snunioo2

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

Step	Hyp	Ref	Expression
1		simp2 1062	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
2		iccid 12220	. . . 4 |- ( B e. RR* -> ( B [,] B ) = { B } )
3	1, 2	syl 17	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( B [,] B ) = { B } )
4	3	uneq2d 3767	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. ( B [,] B ) ) = ( ( A (,) B ) u. { B } ) )
5		simp1 1061	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
6		simp3 1063	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
7		xrleid 11983	. . . 4 |- ( B e. RR* -> B <_ B )
8	1, 7	syl 17	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B <_ B )
9		df-ioo 12179	. . . 4 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
10		df-icc 12182	. . . 4 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
11		xrlenlt 10103	. . . 4 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
12		df-ioc 12180	. . . 4 |- (,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z <_ y ) } )
13		simpl1 1064	. . . . . 6 |- ( ( ( w e. RR* /\ B e. RR* /\ B e. RR* ) /\ ( w < B /\ B <_ B ) ) -> w e. RR* )
14		simpl2 1065	. . . . . 6 |- ( ( ( w e. RR* /\ B e. RR* /\ B e. RR* ) /\ ( w < B /\ B <_ B ) ) -> B e. RR* )
15		simprl 794	. . . . . 6 |- ( ( ( w e. RR* /\ B e. RR* /\ B e. RR* ) /\ ( w < B /\ B <_ B ) ) -> w < B )
16	13, 14, 15	xrltled 39486	. . . . 5 |- ( ( ( w e. RR* /\ B e. RR* /\ B e. RR* ) /\ ( w < B /\ B <_ B ) ) -> w <_ B )
17	16	ex 450	. . . 4 |- ( ( w e. RR* /\ B e. RR* /\ B e. RR* ) -> ( ( w < B /\ B <_ B ) -> w <_ B ) )
18		xrltletr 11988	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( A < B /\ B <_ w ) -> A < w ) )
19	9, 10, 11, 12, 17, 18	ixxun 12191	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ B e. RR* ) /\ ( A < B /\ B <_ B ) ) -> ( ( A (,) B ) u. ( B [,] B ) ) = ( A (,] B ) )
20	5, 1, 1, 6, 8, 19	syl32anc 1334	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. ( B [,] B ) ) = ( A (,] B ) )
21	4, 20	eqtr3d 2658	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   (,]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-ioo 12179  df-ioc 12180  df-icc 12182

This theorem is referenced by:  limcicciooub  39869  limcresiooub  39874  ioccncflimc  40098  volioc  40188  fourierdlem33  40357  fourierdlem49  40372  fourierdlem93  40416  fouriersw  40448