Theorem iccsupr 9961

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 9950	. . . 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 9933	. . . . . . . . 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 4006	. . . . . 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 4002  (class class class)co 5871   RRcr 7806   <_ cle 7988   [,]cicc 9886

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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921

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 4003  df-opab 4064  df-id 4292  df-po 4295  df-iso 4296  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5176  df-fun 5216  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-icc 9890

This theorem is referenced by: (None)