Theorem isposNEW 16773

Description: The predicate "is a poset."

Hypotheses

Ref	Expression
ispos.0NEW	|- S = Struct(2, p, T. )
ispos.1NEW	|- B = (base` K)
ispos.2NEW	|- L = (le` K)

Assertion

Ref	Expression
isposNEW	|- (K e. PosetNEW <-> (K e. S /\ B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))

Distinct variable groups:   x,p,y,z,B   K,p   L,p,x,y,z

Proof of Theorem isposNEW

Step	Hyp	Ref	Expression
1		df-poset 16772	. . 3 |- PosetNEW = Struct(2, p, E.bE.r(b = (base` p) /\ r = (le` p) /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)))))
2	1	eleq2i 1961	. 2 |- (K e. PosetNEW <-> K e. Struct(2, p, E.bE.r(b = (base` p) /\ r = (le` p) /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))))
3		ispos.0NEW	. . 3 |- S = Struct(2, p, T. )
4		fveq2 4681	. . . . . . 7 |- (p = K -> (base` p) = (base` K))
5		ispos.1NEW	. . . . . . 7 |- B = (base` K)
6	4, 5	syl6eqr 1946	. . . . . 6 |- (p = K -> (base` p) = B)
7	6	eqeq2d 1895	. . . . 5 |- (p = K -> (b = (base` p) <-> b = B))
8		fveq2 4681	. . . . . . 7 |- (p = K -> (le` p) = (le` K))
9		ispos.2NEW	. . . . . . 7 |- L = (le` K)
10	8, 9	syl6eqr 1946	. . . . . 6 |- (p = K -> (le` p) = L)
11	10	eqeq2d 1895	. . . . 5 |- (p = K -> (r = (le` p) <-> r = L))
12	7, 11	3anbi12d 1169	. . . 4 |- (p = K -> ((b = (base` p) /\ r = (le` p) /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)))) <-> (b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))))
13	12	2exbidv 1659	. . 3 |- (p = K -> (E.bE.r(b = (base` p) /\ r = (le` p) /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)))) <-> E.bE.r(b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))))
14	3, 13	elstr2 16718	. 2 |- (K e. Struct(2, p, E.bE.r(b = (base` p) /\ r = (le` p) /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))) <-> (K e. S /\ E.bE.r(b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))))
15		fvex 4689	. . . . . 6 |- (base` K) e. _V
16	5, 15	eqeltri 1967	. . . . 5 |- B e. _V
17		fvex 4689	. . . . . 6 |- (le` K) e. _V
18	9, 17	eqeltri 1967	. . . . 5 |- L e. _V
19		neeq1 2024	. . . . . 6 |- (b = B -> (b =/= (/) <-> B =/= (/)))
20		raleq 2266	. . . . . . . 8 |- (b = B -> (A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))
21	20	raleqbi1dv 2271	. . . . . . 7 |- (b = B -> (A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> A.y e. B A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))
22	21	raleqbi1dv 2271	. . . . . 6 |- (b = B -> (A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> A.x e. B A.y e. B A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))
23	19, 22	anbi12d 690	. . . . 5 |- (b = B -> ((b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))) <-> (B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)))))
24		breq 3340	. . . . . . . . 9 |- (r = L -> (xrx <-> xLx))
25		breq 3340	. . . . . . . . . . 11 |- (r = L -> (xry <-> xLy))
26		breq 3340	. . . . . . . . . . 11 |- (r = L -> (yrx <-> yLx))
27	25, 26	anbi12d 690	. . . . . . . . . 10 |- (r = L -> ((xry /\ yrx) <-> (xLy /\ yLx)))
28	27	imbi1d 675	. . . . . . . . 9 |- (r = L -> (((xry /\ yrx) -> x = y) <-> ((xLy /\ yLx) -> x = y)))
29		breq 3340	. . . . . . . . . . 11 |- (r = L -> (yrz <-> yLz))
30	25, 29	anbi12d 690	. . . . . . . . . 10 |- (r = L -> ((xry /\ yrz) <-> (xLy /\ yLz)))
31		breq 3340	. . . . . . . . . 10 |- (r = L -> (xrz <-> xLz))
32	30, 31	imbi12d 688	. . . . . . . . 9 |- (r = L -> (((xry /\ yrz) -> xrz) <-> ((xLy /\ yLz) -> xLz)))
33	24, 28, 32	3anbi123d 1168	. . . . . . . 8 |- (r = L -> ((xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))
34	33	ralbidv 2123	. . . . . . 7 |- (r = L -> (A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))
35	34	2ralbidv 2140	. . . . . 6 |- (r = L -> (A.x e. B A.y e. B A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)) <-> A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))
36	35	anbi2d 678	. . . . 5 |- (r = L -> ((B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))) <-> (B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz)))))
37	16, 18, 23, 36	ceqsex2v 2328	. . . 4 |- (E.bE.r(b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz)))) <-> (B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))
38	37	anbi2i 538	. . 3 |- ((K e. S /\ E.bE.r(b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))) <-> (K e. S /\ (B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz)))))
39		3anass 862	. . 3 |- ((K e. S /\ B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))) <-> (K e. S /\ (B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz)))))
40	38, 39	bitr4i 193	. 2 |- ((K e. S /\ E.bE.r(b = B /\ r = L /\ (b =/= (/) /\ A.x e. b A.y e. b A.z e. b (xrx /\ ((xry /\ yrx) -> x = y) /\ ((xry /\ yrz) -> xrz))))) <-> (K e. S /\ B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))
41	2, 14, 40	3bitri 194	1 |- (K e. PosetNEW <-> (K e. S /\ B =/= (/) /\ A.x e. B A.y e. B A.z e. B (xLx /\ ((xLy /\ yLx) -> x = y) /\ ((xLy /\ yLz) -> xLz))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   T. wtru 1260   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875   class class class wbr 3338  ` cfv 3998  2c2 7145  Structcstru 16707  basecbs 16758  lecple 16759  PosetNEWcpo 16760

This theorem is referenced by:  poslem 16774  isposiNEW 16778

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-tru 1262  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fv 4014  df-opr 4886  df-struct 16708  df-poset 16772

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