Theorem lbicc2 10365

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 1028	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. RR* )
2		xrleid 10181	. . 3 |- ( A e. RR* -> A <_ A )
3	2	3ad2ant1 1049	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ A )
4		simp3 1030	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ B )
5		elicc1 10305	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A e. ( A [,] B ) <-> ( A e. RR* /\ A <_ A /\ A <_ B ) ) )
6	5	3adant3 1048	. 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 1211	1 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RR*cxr 8349   <_ cle 8351   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-icc 10276

This theorem is referenced by:  ivthinclemlm  15658  ivthinclemuopn  15662  ivthdec  15668  cosq34lt1  15874  cos02pilt1  15875