Theorem ubioc1 10125

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 10181. (Contributed by FL, 29-May-2014.)

Assertion

Ref	Expression
ubioc1	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )

Proof of Theorem ubioc1

Step	Hyp	Ref	Expression
1		simp2 1022	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
2		simp3 1023	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
3		xrleid 9996	. . 3 |- ( B e. RR* -> B <_ B )
4	3	3ad2ant2 1043	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B <_ B )
5		elioc1 10118	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
6	5	3adant3 1041	. 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 1204	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RR*cxr 8180   < clt 8181   <_ cle 8182   (,]cioc 10085

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-ioc 10089

This theorem is referenced by: (None)