Theorem supmaxOLD 8014

Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.) Obsolete version of supmax 8012 as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
supmaxOLD.1	|- ( ph -> R Or A )
supmaxOLD.2	|- ( ph -> C e. A )
supmaxOLD.3	|- ( ph -> C e. B )
supmaxOLD.4	|- ( ( ph /\ y e. B ) -> -. C R y )

Assertion

Ref	Expression
supmaxOLD	|- ( ph -> sup ( B , A , R ) = C )

Distinct variable groups:   y, A   y, B   y, C   y, R   ph, y

Proof of Theorem supmaxOLD

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmaxOLD.3	. . 3 |- ( ph -> C e. B )
2		supmaxOLD.1	. . . 4 |- ( ph -> R Or A )
3		supmaxOLD.2	. . . . 5 |- ( ph -> C e. A )
4		supmaxOLD.4	. . . . . 6 |- ( ( ph /\ y e. B ) -> -. C R y )
5	4	ralrimiva 2814	. . . . 5 |- ( ph -> A. y e. B -. C R y )
6		supmaxlemOLD 8013	. . . . 5 |- ( ( C e. A /\ C e. B /\ A. y e. B -. C R y ) -> E. x e. A ( A. z e. B -. x R z /\ A. z e. A ( z R x -> E. y e. B z R y ) ) )
7	3, 1, 5, 6	syl3anc 1276	. . . 4 |- ( ph -> E. x e. A ( A. z e. B -. x R z /\ A. z e. A ( z R x -> E. y e. B z R y ) ) )
8	2, 7	supub 8004	. . 3 |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) )
9	1, 8	mpd 15	. 2 |- ( ph -> -. sup ( B , A , R ) R C )
10	2, 7	supnub 8007	. . 3 |- ( ph -> ( ( C e. A /\ A. y e. B -. C R y ) -> -. C R sup ( B , A , R ) ) )
11	3, 5, 10	mp2and 690	. 2 |- ( ph -> -. C R sup ( B , A , R ) )
12	2, 7	supcl 8003	. . 3 |- ( ph -> sup ( B , A , R ) e. A )
13		sotrieq2 4805	. . 3 |- ( ( R Or A /\ ( sup ( B , A , R ) e. A /\ C e. A ) ) -> ( sup ( B , A , R ) = C <-> ( -. sup ( B , A , R ) R C /\ -. C R sup ( B , A , R ) ) ) )
14	2, 12, 3, 13	syl12anc 1274	. 2 |- ( ph -> ( sup ( B , A , R ) = C <-> ( -. sup ( B , A , R ) R C /\ -. C R sup ( B , A , R ) ) ) )
15	9, 11, 14	mpbir2and 938	1 |- ( ph -> sup ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   A.wral 2749   E.wrex 2750   class class class wbr 4418   Or wor 4776   supcsup 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-po 4777  df-so 4778  df-iota 5569  df-riota 6282  df-sup 7987

This theorem is referenced by: (None)