Theorem iccsupr 10299

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 10288	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
2		sstr 3246	. . . . 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 1044	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S C_ RR )
6		ne0i 3515	. . 3 |- ( C e. S -> S =/= (/) )
7	6	3ad2ant3 1047	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) /\ C e. S ) -> S =/= (/) )
8		simplr 529	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> B e. RR )
9		ssel 3232	. . . . . . . 8 |- ( S C_ ( A [,] B ) -> ( y e. S -> y e. ( A [,] B ) ) )
10		elicc2 10271	. . . . . . . . 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 1038	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) /\ y e. S ) -> y <_ B )
15	14	ralrimiva 2615	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ S C_ ( A [,] B ) ) -> A. y e. S y <_ B )
16		breq2 4113	. . . . . 6 |- ( x = B -> ( y <_ x <-> y <_ B ) )
17	16	ralbidv 2542	. . . . 5 |- ( x = B -> ( A. y e. S y <_ x <-> A. y e. S y <_ B ) )
18	17	rspcev 2921	. . . 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 1044	. 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 1204	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 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   E.wrex 2521   C_ wss 3211   (/)c0 3508   class class class wbr 4109  (class class class)co 6050   RRcr 8126   <_ cle 8309   [,]cicc 10224

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-icc 10228

This theorem is referenced by: (None)