Theorem fisupg 15748

Description: Lemma showing existence and closure of supremum of a finite set.

Assertion

Ref	Expression
fisupg	|- ((R Or A /\ A e. Fin /\ A =/= (/)) -> E.x e. A (A.y e. A -. xRy /\ A.y e. A (yRx -> E.z e. A yRz)))

Distinct variable groups:   x,R,y,z   x,A,y,z

Proof of Theorem fisupg

Step	Hyp	Ref	Expression
1		fimaxg 15747	. 2 |- ((R Or A /\ A e. Fin /\ A =/= (/)) -> E.x e. A A.y e. A (x =/= y -> yRx))
2		sotrieq2 3618	. . . . . . . . . . 11 |- ((R Or A /\ (x e. A /\ y e. A)) -> (x = y <-> (-. xRy /\ -. yRx)))
3	2	simprbda 464	. . . . . . . . . 10 |- (((R Or A /\ (x e. A /\ y e. A)) /\ x = y) -> -. xRy)
4	3	ex 402	. . . . . . . . 9 |- ((R Or A /\ (x e. A /\ y e. A)) -> (x = y -> -. xRy))
5	4	anassrs 489	. . . . . . . 8 |- (((R Or A /\ x e. A) /\ y e. A) -> (x = y -> -. xRy))
6	5	a1dd 53	. . . . . . 7 |- (((R Or A /\ x e. A) /\ y e. A) -> (x = y -> ((x =/= y -> yRx) -> -. xRy)))
7		pm2.27 76	. . . . . . . 8 |- (x =/= y -> ((x =/= y -> yRx) -> yRx))
8		pm3.21 306	. . . . . . . . . . 11 |- (yRx -> (xRy -> (xRy /\ yRx)))
9	8	con3d 111	. . . . . . . . . 10 |- (yRx -> (-. (xRy /\ yRx) -> -. xRy))
10		so2nr 3613	. . . . . . . . . 10 |- ((R Or A /\ (x e. A /\ y e. A)) -> -. (xRy /\ yRx))
11	9, 10	syl5com 63	. . . . . . . . 9 |- ((R Or A /\ (x e. A /\ y e. A)) -> (yRx -> -. xRy))
12	11	anassrs 489	. . . . . . . 8 |- (((R Or A /\ x e. A) /\ y e. A) -> (yRx -> -. xRy))
13	7, 12	syl9r 72	. . . . . . 7 |- (((R Or A /\ x e. A) /\ y e. A) -> (x =/= y -> ((x =/= y -> yRx) -> -. xRy)))
14	6, 13	pm2.61dne 2091	. . . . . 6 |- (((R Or A /\ x e. A) /\ y e. A) -> ((x =/= y -> yRx) -> -. xRy))
15	14	ralimdvaa 2171	. . . . 5 |- ((R Or A /\ x e. A) -> (A.y e. A (x =/= y -> yRx) -> A.y e. A -. xRy))
16		breq2 3342	. . . . . . . . . 10 |- (z = x -> (yRz <-> yRx))
17	16	rcla4ev 2381	. . . . . . . . 9 |- ((x e. A /\ yRx) -> E.z e. A yRz)
18	17	ex 402	. . . . . . . 8 |- (x e. A -> (yRx -> E.z e. A yRz))
19	18	adantr 425	. . . . . . 7 |- ((x e. A /\ y e. A) -> (yRx -> E.z e. A yRz))
20	19	r19.21aiva 2176	. . . . . 6 |- (x e. A -> A.y e. A (yRx -> E.z e. A yRz))
21	20	adantl 424	. . . . 5 |- ((R Or A /\ x e. A) -> A.y e. A (yRx -> E.z e. A yRz))
22	15, 21	jctird 663	. . . 4 |- ((R Or A /\ x e. A) -> (A.y e. A (x =/= y -> yRx) -> (A.y e. A -. xRy /\ A.y e. A (yRx -> E.z e. A yRz))))
23	22	reximdva 2203	. . 3 |- (R Or A -> (E.x e. A A.y e. A (x =/= y -> yRx) -> E.x e. A (A.y e. A -. xRy /\ A.y e. A (yRx -> E.z e. A yRz))))
24	23	3ad2ant1 897	. 2 |- ((R Or A /\ A e. Fin /\ A =/= (/)) -> (E.x e. A A.y e. A (x =/= y -> yRx) -> E.x e. A (A.y e. A -. xRy /\ A.y e. A (yRx -> E.z e. A yRz))))
25	1, 24	mpd 29	1 |- ((R Or A /\ A e. Fin /\ A =/= (/)) -> E.x e. A (A.y e. A -. xRy /\ A.y e. A (yRx -> E.z e. A yRz)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  (/)c0 2875   class class class wbr 3338   Or wor 3590  Fincfn 5426

This theorem is referenced by:  fisup2g 15768

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-3or 859  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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-1o 5177  df-er 5318  df-en 5427  df-fin 5430

Copyright terms: Public domain