Theorem ubioc1 9898

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 9954. (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 998	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* )
2		simp3 999	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
3		xrleid 9769	. . 3 |- ( B e. RR* -> B <_ B )
4	3	3ad2ant2 1019	. 2 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B <_ B )
5		elioc1 9891	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B e. ( A (,] B ) <-> ( B e. RR* /\ A < B /\ B <_ B ) ) )
6	5	3adant3 1017	. 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 1180	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. ( A (,] B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   e. wcel 2146   class class class wbr 3998  (class class class)co 5865   RR*cxr 7965   < clt 7966   <_ cle 7967   (,]cioc 9858

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-pre-ltirr 7898

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-ioc 9862

This theorem is referenced by: (None)