Theorem iccsupr 8835

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 8824	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
2		sstr 2953	. . . . 5 |- ( ( S C_ ( A [,] B ) /\ ( A [,] B ) C_ RR ) -> S C_ RR )
3	2	ancoms 255	. . . 4 |- ( ( ( A [,] B ) C_ RR /\ S C_ ( A [,] B ) ) -> S C_ RR )
4	1, 3	sylan 267	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> S C_ RR )
5	4	3adant3 924	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S C_ RR )
6		ne0i 3230	. . 3 |- ( C e. S -> S =/= (/) )
7	6	3ad2ant3 927	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S =/= (/) )
8		simplr 482	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> B e. RR )
9		ssel 2939	. . . . . . . 8 |- ( S C_ ( A [,] B ) -> ( y e. S -> y e. ( A [,] B ) ) )
10		elicc2 8807	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) <-> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
11	10	biimpd 132	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( y e. ( A [,] B ) -> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
12	9, 11	sylan9r 390	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> ( y e. S -> ( y e. RR /\ A <_ y /\ y <_ B ) ) )
13	12	imp 115	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> ( y e. RR /\ A <_ y /\ y <_ B ) )
14	13	simp3d 918	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> y <_ B )
15	14	ralrimiva 2392	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> A. y e. S y <_ B )
16		breq2 3768	. . . . . 6 |- ( x = B -> ( y <_ x <-> y <_ B ) )
17	16	ralbidv 2326	. . . . 5 |- ( x = B -> ( A. y e. S y <_ x <-> A. y e. S y <_ B ) )
18	17	rspcev 2656	. . . 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 391	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> E. x e. RR A. y e. S y <_ x )
20	19	3adant3 924	. 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 1084	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393   =/= wne 2204   A.wral 2306   E.wrex 2307   C_ wss 2917   (/)c0 3224   class class class wbr 3764  (class class class)co 5512   RRcr 6888   <_ cle 7061   [,]cicc 8760

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-icc 8764

This theorem is referenced by: (None)