Theorem ubicc2 10219

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

Assertion

Ref	Expression
ubicc2	|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )

Proof of Theorem ubicc2

Step	Hyp	Ref	Expression
1		simp2 1024	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. RR* )
2		simp3 1025	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A <_ B )
3		xrleid 10034	. . 3 |- ( B e. RR* -> B <_ B )
4	3	3ad2ant2 1045	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B <_ B )
5		elicc1 10158	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B e. ( A [,] B ) <-> ( B e. RR* /\ A <_ B /\ B <_ B ) ) )
6	5	3adant3 1043	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( B e. ( A [,] B ) <-> ( B e. RR* /\ A <_ B /\ B <_ B ) ) )
7	1, 2, 4, 6	mpbir3and 1206	1 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RR*cxr 8212   <_ cle 8214   [,]cicc 10125

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-icc 10129

This theorem is referenced by:  ivthinclemum  15358  ivthinclemlopn  15359  ivthdec  15367  cos0pilt1  15575