Theorem iccsupr 9964

Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)

Assertion

Ref	Expression
iccsupr	|- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> ( S C_ RR /\ S =/= (/) /\ E. x e. RR A. y e. S y <_ x ) )

Distinct variable groups:   y, A   x, B, y   x, S, y

Allowed substitution hints:   A( x)   C( x, y)

Proof of Theorem iccsupr

Step	Hyp	Ref	Expression
1		iccssre 9953	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
2		sstr 3163	. . . . 5 |- ( ( S C_ ( A [,] B ) /\ ( A [,] B ) C_ RR ) -> S C_ RR )
3	2	ancoms 268	. . . 4 |- ( ( ( A [,] B ) C_ RR /\ S C_ ( A [,] B ) ) -> S C_ RR )
4	1, 3	sylan 283	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> S C_ RR )
5	4	3adant3 1017	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S C_ RR )
6		ne0i 3429	. . 3 |- ( C e. S -> S =/= (/) )
7	6	3ad2ant3 1020	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S =/= (/) )
8		simplr 528	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> B e. RR )
9		ssel 3149	. . . . . . . 8 |- ( S C_ ( A [,] B ) -> ( y e. S -> y e. ( A [,] B ) ) )
10		elicc2 9936	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
11	10	biimpd 144	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) -> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
12	9, 11	sylan9r 410	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> ( y e. S -> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
13	12	imp 124	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> ( y e. RR /\ A <_ y /\ y <_ B ) )
14	13	simp3d 1011	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> y <_ B )
15	14	ralrimiva 2550	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> A. y e. S y <_ B )
16		breq2 4007	. . . . . 6 |- ( x = B -> ( y <_ x <-> y <_ B ) )
17	16	ralbidv 2477	. . . . 5 |- ( x = B -> ( A. y e. S y <_ x <-> A. y e. S y <_ B ) )
18	17	rspcev 2841	. . . 4 |- ( ( B e. RR /\ A. y e. S y <_ B ) -> E. x e. RR A. y e. S y <_ x )
19	8, 15, 18	syl2anc 411	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> E. x e. RR A. y e. S y <_ x )
20	19	3adant3 1017	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> E. x e. RR A. y e. S y <_ x )
21	5, 7, 20	3jca 1177	1 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> ( S C_ RR /\ S =/= (/) /\ E. x e. RR A. y e. S y <_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   E.wrex 2456   C_ wss 3129   (/)c0 3422   class class class wbr 4003  (class class class)co 5874   RRcr 7809   <_ cle 7991   [,]cicc 9889

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-icc 9893

This theorem is referenced by: (None)