Theorem supmaxlemOLD 7945

Description: A set that contains the greatest element satisfies the antecedent in supremum theorems. This allows to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.) Obsolete as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
supmaxlemOLD

Distinct variable groups:   ,   ,,,   ,,,   ,,,

Proof of Theorem supmaxlemOLD

Step	Hyp	Ref	Expression
1		breq2 4456	. . . . . . 7
2	1	rspcev 3210	. . . . . 6
3	2	ex 434	. . . . 5
4	3	ralrimivw 2872	. . . 4
5		breq2 4456	. . . . . . 7
6	5	notbid 294	. . . . . 6
7	6	cbvralv 3084	. . . . 5
8	7	biimpi 194	. . . 4
9	4, 8	anim12ci 567	. . 3
10		breq1 4455	. . . . . . 7
11	10	notbid 294	. . . . . 6
12	11	ralbidv 2896	. . . . 5
13		breq2 4456	. . . . . . 7
14	13	imbi1d 317	. . . . . 6
15	14	ralbidv 2896	. . . . 5
16	12, 15	anbi12d 710	. . . 4
17	16	rspcev 3210	. . 3
18	9, 17	sylan2 474	. 2
19	18	3impb 1192	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   class class class wbr 4452

This theorem is referenced by:  supmaxOLD  7946

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-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  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-br 4453