Theorem weinxp 4059

Description: Intersection of well-ordering with cross product of its field.

Assertion

Ref	Expression
weinxp	|- (R We A <-> (R i^i (A X. A)) We A)

Proof of Theorem weinxp

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . . . . . . . 14 |- (z C_ A -> (x e. z -> x e. A))
2		ssel 2615	. . . . . . . . . . . . . 14 |- (z C_ A -> (y e. z -> y e. A))
3	1, 2	anim12d 617	. . . . . . . . . . . . 13 |- (z C_ A -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
4		brinxp 4058	. . . . . . . . . . . . . 14 |- ((y e. A /\ x e. A) -> (yRx <-> y(R i^i (A X. A))x))
5	4	ancoms 484	. . . . . . . . . . . . 13 |- ((x e. A /\ y e. A) -> (yRx <-> y(R i^i (A X. A))x))
6	3, 5	syl6 25	. . . . . . . . . . . 12 |- (z C_ A -> ((x e. z /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x)))
7	6	exp3a 405	. . . . . . . . . . 11 |- (z C_ A -> (x e. z -> (y e. z -> (yRx <-> y(R i^i (A X. A))x))))
8	7	imp31 389	. . . . . . . . . 10 |- (((z C_ A /\ x e. z) /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x))
9	8	notbid 673	. . . . . . . . 9 |- (((z C_ A /\ x e. z) /\ y e. z) -> (-. yRx <-> -. y(R i^i (A X. A))x))
10	9	ralbidva 2119	. . . . . . . 8 |- ((z C_ A /\ x e. z) -> (A.y e. z -. yRx <-> A.y e. z -. y(R i^i (A X. A))x))
11	10	rexbidva 2120	. . . . . . 7 |- (z C_ A -> (E.x e. z A.y e. z -. yRx <-> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
12	11	adantr 425	. . . . . 6 |- ((z C_ A /\ z =/= (/)) -> (E.x e. z A.y e. z -. yRx <-> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
13	12	pm5.74i 644	. . . . 5 |- (((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> ((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
14	13	albii 1346	. . . 4 |- (A.z((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> A.z((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
15		df-fr 3625	. . . 4 |- (R Fr A <-> A.z((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx))
16		df-fr 3625	. . . 4 |- ((R i^i (A X. A)) Fr A <-> A.z((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
17	14, 15, 16	3bitr4i 200	. . 3 |- (R Fr A <-> (R i^i (A X. A)) Fr A)
18		brinxp 4058	. . . . . . 7 |- ((x e. A /\ y e. A) -> (xRy <-> x(R i^i (A X. A))y))
19		biidd 188	. . . . . . 7 |- ((x e. A /\ y e. A) -> (x = y <-> x = y))
20	18, 19, 5	3orbi123d 1167	. . . . . 6 |- ((x e. A /\ y e. A) -> ((xRy \/ x = y \/ yRx) <-> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
21	20	pm5.74i 644	. . . . 5 |- (((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)) <-> ((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
22	21	2albii 1347	. . . 4 |- (A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)) <-> A.xA.y((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
23		r2al 2136	. . . 4 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
24		r2al 2136	. . . 4 |- (A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x) <-> A.xA.y((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
25	22, 23, 24	3bitr4i 200	. . 3 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x))
26	17, 25	anbi12i 540	. 2 |- ((R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) <-> ((R i^i (A X. A)) Fr A /\ A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
27		dfwe2 3861	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
28		dfwe2 3861	. 2 |- ((R i^i (A X. A)) We A <-> ((R i^i (A X. A)) Fr A /\ A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
29	26, 27, 28	3bitr4i 200	1 |- (R We A <-> (R i^i (A X. A)) We A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857  A.wal 1296   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338   Fr wfr 3623   We wwe 3624   X. cxp 3984

This theorem is referenced by:  weth 5949

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-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-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-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-xp 4000

Copyright terms: Public domain