Theorem lbicc2 10320

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 1024	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. RR* )
2		xrleid 10136	. . 3 |- ( A e. RR* -> A <_ A )
3	2	3ad2ant1 1045	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ A )
4		simp3 1026	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ B )
5		elicc1 10260	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A e. ( A [,] B ) <-> ( A e. RR* /\ A <_ A /\ A <_ B ) ) )
6	5	3adant3 1044	. 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 1207	1 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   RR*cxr 8309   <_ cle 8311   [,]cicc 10227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-pre-ltirr 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-icc 10231

This theorem is referenced by:  ivthinclemlm  15516  ivthinclemuopn  15520  ivthdec  15526  cosq34lt1  15732  cos02pilt1  15733