Theorem iccssre 10180

Description: A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)

Assertion

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

Proof of Theorem iccssre

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elicc2 10163	. . . . 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 3231	1 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   C_ wss 3198   class class class wbr 4086  (class class class)co 6013   RRcr 8021   <_ cle 8205   [,]cicc 10116

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-icc 10120

This theorem is referenced by:  iccsupr  10191  iccshftri  10220  iccshftli  10222  iccdili  10224  icccntri  10226  unitssre  10230  iccen  10231  cos12dec  12319  suplociccreex  15338  suplociccex  15339  dedekindicclemuub  15340  dedekindicclemlu  15344  dedekindicclemeu  15345  dedekindicclemicc  15346  dedekindicc  15347  reeff1olem  15485  cosz12  15494  ioocosf1o  15568