Theorem snunioo 11416

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 988	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
2		iccid 11350	. . . 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 3514	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( { A } u. ( A (,) B ) ) )
5		simp2 989	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
6		xrleid 11132	. . . 4 |- ( A e. RR* -> A <_ A )
7	1, 6	syl 16	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A )
8		simp3 990	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
9		df-icc 11312	. . . 4 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
10		df-ioo 11309	. . . 4 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
11		xrltnle 9448	. . . 4 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) )
12		df-ico 11311	. . . 4 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
13		xrlelttr 11135	. . . 4 |- ( ( w e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( w <_ A /\ A < B ) -> w < B ) )
14		xrltle 11131	. . . . . 6 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) )
15	14	3adant1 1006	. . . . 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 11321	. . 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 1226	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) )
19	4, 18	eqtr3d 2477	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )

Syntax hints:   -> wi 4   /\ w3a 965   = wceq 1369   e. wcel 1756   u. cun 3331   {csn 3882   class class class wbr 4297  (class class class)co 6096   RR*cxr 9422   < clt 9423   <_ cle 9424   (,)cioo 11305   [,)cico 11307   [,]cicc 11308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-ioo 11309  df-ico 11311  df-icc 11312

This theorem is referenced by:  prunioo  11419  ioojoin  11421  icombl1  21049  ioombl  21051  tan2h  28429  mbfposadd  28444  itg2addnclem2  28449  ftc1anclem5  28476  iocunico  29591