Theorem supmax 8373

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.) (Proof shortened by OpenAI, 30-Mar-2020.)

Hypotheses

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

Assertion

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

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

Proof of Theorem supmax

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmax.1	. 2 |- ( ph -> R Or A )
2		supmax.2	. 2 |- ( ph -> C e. A )
3		supmax.4	. 2 |- ( ( ph /\ y e. B ) -> -. C R y )
4		supmax.3	. . 3 |- ( ph -> C e. B )
5		simprr 796	. . 3 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> y R C )
6		breq2 4657	. . . 4 |- ( z = C -> ( y R z <-> y R C ) )
7	6	rspcev 3309	. . 3 |- ( ( C e. B /\ y R C ) -> E. z e. B y R z )
8	4, 5, 7	syl2an2r 876	. 2 |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. z e. B y R z )
9	1, 2, 3, 8	eqsupd 8363	1 |- ( ph -> sup ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   Or wor 5034   supcsup 8346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-iota 5851  df-riota 6611  df-sup 8348

This theorem is referenced by:  suppr  8377  gsumesum  30121  supfz  31613  inffzOLD  31615  mblfinlem2  33447