Theorem iocssre 10145

Description: A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)

Assertion

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

Proof of Theorem iocssre

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elioc2 10128	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
2	1	biimp3a 1379	. . . 4 |- ( ( A e. RR* /\ B e. RR /\ x e. ( A (,] B ) ) -> ( x e. RR /\ A < x /\ x <_ B ) )
3	2	simp1d 1033	. . 3 |- ( ( A e. RR* /\ B e. RR /\ x e. ( A (,] B ) ) -> x e. RR )
4	3	3expia 1229	. 2 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) -> x e. RR ) )
5	4	ssrdv 3230	1 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   C_ wss 3197   class class class wbr 4082  (class class class)co 6000   RRcr 7994   RR*cxr 8176   < clt 8177   <_ cle 8178   (,]cioc 10081

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109

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 3888  df-br 4083  df-opab 4145  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-ioc 10085

This theorem is referenced by:  negpitopissre  15523