Theorem lbicc2 11639

Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)

Assertion

Ref	Expression
lbicc2	|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )

Proof of Theorem lbicc2

Step	Hyp	Ref	Expression
1		simp1 994	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. RR* )
2		xrleid 11359	. . 3 |- ( A e. RR* -> A <_ A )
3	2	3ad2ant1 1015	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ A )
4		simp3 996	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ B )
5		elicc1 11576	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A e. ( A [,] B ) <-> ( A e. RR* /\ A <_ A /\ A <_ B ) ) )
6	5	3adant3 1014	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( A e. ( A [,] B ) <-> ( A e. RR* /\ A <_ A /\ A <_ B ) ) )
7	1, 3, 4, 6	mpbir3and 1177	1 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 971   e. wcel 1823   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616   <_ cle 9618   [,]cicc 11535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-icc 11539

This theorem is referenced by:  icccmplem1  21493  reconnlem2  21498  oprpiece1res1  21617  pcoass  21690  ivthlem1  22029  ivth2  22033  ivthle  22034  ivthle2  22035  evthicc  22037  ovolicc2lem5  22098  dyadmaxlem  22172  rolle  22557  cmvth  22558  mvth  22559  dvlip  22560  c1liplem1  22563  dveq0  22567  dvgt0lem1  22569  lhop1lem  22580  dvcnvrelem1  22584  dvcvx  22587  dvfsumle  22588  dvfsumge  22589  dvfsumabs  22590  dvfsumlem2  22594  ftc2  22611  ftc2ditglem  22612  itgparts  22614  itgsubstlem  22615  taylfval  22920  tayl0  22923  efcvx  23010  pige3  23076  logccv  23212  loglesqrt  23300  eliccioo  27861  oms0  28505  cvmliftlem6  28999  cvmliftlem8  29001  cvmliftlem9  29002  cvmliftlem10  29003  cvmliftlem13  29005  ftc2nc  30339  areacirc  30352  ivthALT  30393  itgpowd  31423  iccintsng  31802  icccncfext  31929  cncfiooicclem1  31935  dvbdfbdioolem1  31964  itgsin0pilem1  31987  itgcoscmulx  32007  itgsincmulx  32012  fourierdlem20  32148  fourierdlem51  32179  fourierdlem54  32182  fourierdlem64  32192  fourierdlem73  32201  fourierdlem81  32209  fourierdlem102  32230  fourierdlem103  32231  fourierdlem104  32232  fourierdlem114  32242  etransclem46  32302