Theorem supmaxOLD 7946

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 7944 as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
supmaxOLD.1
supmaxOLD.2
supmaxOLD.3
supmaxOLD.4

Assertion

Ref	Expression
supmaxOLD

Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem supmaxOLD

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

Step	Hyp	Ref	Expression
1		supmaxOLD.3	. . 3
2		supmaxOLD.1	. . . 4
3		supmaxOLD.2	. . . . 5
4		supmaxOLD.4	. . . . . 6
5	4	ralrimiva 2871	. . . . 5
6		supmaxlemOLD 7945	. . . . 5
7	3, 1, 5, 6	syl3anc 1228	. . . 4
8	2, 7	supub 7939	. . 3
9	1, 8	mpd 15	. 2
10	2, 7	supnub 7942	. . 3
11	3, 5, 10	mp2and 679	. 2
12	2, 7	supcl 7938	. . 3
13		sotrieq2 4833	. . 3
14	2, 12, 3, 13	syl12anc 1226	. 2
15	9, 11, 14	mpbir2and 922	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   class class class wbr 4452  Orwor 4804  supcsup 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-iota 5556  df-riota 6257  df-sup 7921