Theorem geme2 14617

Description: The greatest element of X is a maximal element.

Hypothesis

Ref	Expression
geme2.1	|- X = dom R

Assertion

Ref	Expression
geme2	|- ((R e. Poset /\ (R supw X) e. X) -> (R supw X) e. (mxl` R))

Proof of Theorem geme2

Step	Hyp	Ref	Expression
1		geme2.1	. . . . . 6 |- X = dom R
2	1	eleq2i 1961	. . . . 5 |- ((R supw X) e. X <-> (R supw X) e. dom R)
3	2	biimpi 168	. . . 4 |- ((R supw X) e. X -> (R supw X) e. dom R)
4	3	adantl 424	. . 3 |- ((R e. Poset /\ (R supw X) e. X) -> (R supw X) e. dom R)
5		simpll 448	. . . . 5 |- (((R e. Poset /\ (R supw X) e. X) /\ x e. dom R) -> R e. Poset)
6		simpr 350	. . . . . 6 |- (((R e. Poset /\ (R supw X) e. X) /\ x e. dom R) -> x e. dom R)
7	1	opreq2i 4893	. . . . . . . . 9 |- (R supw X) = (R supw dom R)
8	7, 1	eleq12i 1962	. . . . . . . 8 |- ((R supw X) e. X <-> (R supw dom R) e. dom R)
9	8	biimpi 168	. . . . . . 7 |- ((R supw X) e. X -> (R supw dom R) e. dom R)
10	9	ad2antlr 441	. . . . . 6 |- (((R e. Poset /\ (R supw X) e. X) /\ x e. dom R) -> (R supw dom R) e. dom R)
11		eqid 1884	. . . . . . 7 |- dom R = dom R
12	11	supdefa 14605	. . . . . 6 |- ((R e. Poset /\ x e. dom R /\ (R supw dom R) e. dom R) -> xR(R supw dom R))
13	5, 6, 10, 12	syl111anc 1100	. . . . 5 |- (((R e. Poset /\ (R supw X) e. X) /\ x e. dom R) -> xR(R supw dom R))
14		opreq2 4890	. . . . . . . . . . . 12 |- (dom R = X -> (R supw dom R) = (R supw X))
15	14	breq2d 3350	. . . . . . . . . . 11 |- (dom R = X -> (xR(R supw dom R) <-> xR(R supw X)))
16	15	3anbi3d 1174	. . . . . . . . . 10 |- (dom R = X -> ((R e. Poset /\ (R supw X)Rx /\ xR(R supw dom R)) <-> (R e. Poset /\ (R supw X)Rx /\ xR(R supw X))))
17		psasym 9994	. . . . . . . . . 10 |- ((R e. Poset /\ (R supw X)Rx /\ xR(R supw X)) -> (R supw X) = x)
18	16, 17	syl6bi 231	. . . . . . . . 9 |- (dom R = X -> ((R e. Poset /\ (R supw X)Rx /\ xR(R supw dom R)) -> (R supw X) = x))
19	18	eqcoms 1887	. . . . . . . 8 |- (X = dom R -> ((R e. Poset /\ (R supw X)Rx /\ xR(R supw dom R)) -> (R supw X) = x))
20	1, 19	ax-mp 7	. . . . . . 7 |- ((R e. Poset /\ (R supw X)Rx /\ xR(R supw dom R)) -> (R supw X) = x)
21	20	3exp 1066	. . . . . 6 |- (R e. Poset -> ((R supw X)Rx -> (xR(R supw dom R) -> (R supw X) = x)))
22	21	com23 36	. . . . 5 |- (R e. Poset -> (xR(R supw dom R) -> ((R supw X)Rx -> (R supw X) = x)))
23	5, 13, 22	sylc 83	. . . 4 |- (((R e. Poset /\ (R supw X) e. X) /\ x e. dom R) -> ((R supw X)Rx -> (R supw X) = x))
24	23	r19.21aiva 2176	. . 3 |- ((R e. Poset /\ (R supw X) e. X) -> A.x e. dom R((R supw X)Rx -> (R supw X) = x))
25	4, 24	jca 310	. 2 |- ((R e. Poset /\ (R supw X) e. X) -> ((R supw X) e. dom R /\ A.x e. dom R((R supw X)Rx -> (R supw X) = x)))
26		posispre 14582	. . . 4 |- (R e. Poset -> R e. Preset )
27	11	ismxl2 14609	. . . 4 |- (R e. Preset -> ((R supw X) e. (mxl` R) <-> ((R supw X) e. dom R /\ A.x e. dom R((R supw X)Rx -> (R supw X) = x))))
28	26, 27	syl 12	. . 3 |- (R e. Poset -> ((R supw X) e. (mxl` R) <-> ((R supw X) e. dom R /\ A.x e. dom R((R supw X)Rx -> (R supw X) = x))))
29	28	adantr 425	. 2 |- ((R e. Poset /\ (R supw X) e. X) -> ((R supw X) e. (mxl` R) <-> ((R supw X) e. dom R /\ A.x e. dom R((R supw X)Rx -> (R supw X) = x))))
30	25, 29	mpbird 213	1 |- ((R e. Poset /\ (R supw X) e. X) -> (R supw X) e. (mxl` R))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  dom cdm 3986  ` cfv 3998  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981   Preset cpreset 14555  mxlcmxl 14556

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-ps 9984  df-spw 9985  df-prs 14563  df-mxl 14589

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