Theorem iccsupr 10347

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 10336	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
2		sstr 3256	. . . . 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 1048	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S C_ RR )
6		ne0i 3528	. . 3 |- ( C e. S -> S =/= (/) )
7	6	3ad2ant3 1051	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S =/= (/) )
8		simplr 533	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> B e. RR )
9		ssel 3242	. . . . . . . 8 |- ( S C_ ( A [,] B ) -> ( y e. S -> y e. ( A [,] B ) ) )
10		elicc2 10319	. . . . . . . . 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 414	. . . . . . 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 1042	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> y <_ B )
15	14	ralrimiva 2623	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> A. y e. S y <_ B )
16		breq2 4129	. . . . . 6 |- ( x = B -> ( y <_ x <-> y <_ B ) )
17	16	ralbidv 2550	. . . . 5 |- ( x = B -> ( A. y e. S y <_ x <-> A. y e. S y <_ B ) )
18	17	rspcev 2929	. . . 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 415	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> E. x e. RR A. y e. S y <_ x )
20	19	3adant3 1048	. 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 1208	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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   C_ wss 3220   (/)c0 3520   class class class wbr 4125  (class class class)co 6075   RRcr 8168   <_ cle 8351   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-icc 10276

This theorem is referenced by: (None)