Theorem weinxp 4192

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 2846	. . . . . . . . . . . . . 14 |- (z C_ A -> (x e. z -> x e. A))
2		ssel 2846	. . . . . . . . . . . . . 14 |- (z C_ A -> (y e. z -> y e. A))
3	1, 2	anim12d 533	. . . . . . . . . . . . 13 |- (z C_ A -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
4		brinxp 4191	. . . . . . . . . . . . . 14 |- ((y e. A /\ x e. A) -> (yRx <-> y(R i^i (A X. A))x))
5	4	ancoms 416	. . . . . . . . . . . . 13 |- ((x e. A /\ y e. A) -> (yRx <-> y(R i^i (A X. A))x))
6	3, 5	syl6 42	. . . . . . . . . . . 12 |- (z C_ A -> ((x e. z /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x)))
7	6	exp3a 400	. . . . . . . . . . 11 |- (z C_ A -> (x e. z -> (y e. z -> (yRx <-> y(R i^i (A X. A))x))))
8	7	imp31 396	. . . . . . . . . 10 |- (((z C_ A /\ x e. z) /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x))
9	8	notbid 746	. . . . . . . . 9 |- (((z C_ A /\ x e. z) /\ y e. z) -> (-. yRx <-> -. y(R i^i (A X. A))x))
10	9	ralbidva 2369	. . . . . . . 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 2370	. . . . . . 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 447	. . . . . 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 275	. . . . 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 1635	. . . 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 3782	. . . 4 |- (R Fr A <-> A.z((z C_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx))
16		df-fr 3782	. . . 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 295	. . 3 |- (R Fr A <-> (R i^i (A X. A)) Fr A)
18		brinxp 4191	. . . . . . 7 |- ((x e. A /\ y e. A) -> (xRy <-> x(R i^i (A X. A))y))
19		biidd 241	. . . . . . 7 |- ((x e. A /\ y e. A) -> (x = y <-> x = y))
20	18, 19, 5	3orbi123d 1440	. . . . . 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 275	. . . . 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 1636	. . . 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 2386	. . . 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 2386	. . . 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 295	. . 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 710	. 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 4007	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
28		dfwe2 4007	. 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 295	1 |- (R We A <-> (R i^i (A X. A)) We A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 219   /\ wa 337   \/ w3o 1101  A.wal 1584   = wceq 1586   e. wcel 1588   =/= wne 2266  A.wral 2355  E.wrex 2356   i^i cin 2826   C_ wss 2827  (/)c0 3082   class class class wbr 3507   Fr wfr 3780   We wwe 3781   X. cxp 4117

This theorem is referenced by:  infxpenlem 6147  aceq8b 6200  ac10ct 6204  wethOLD 6360

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-13 1599  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-pr 3687  ax-un 3929

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-3or 1103  df-3an 1104  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-v 2540  df-dif 2830  df-un 2832  df-in 2834  df-ss 2836  df-nul 3083  df-pw 3229  df-sn 3242  df-pr 3243  df-tp 3245  df-op 3246  df-uni 3367  df-br 3508  df-opab 3566  df-po 3752  df-so 3764  df-fr 3782  df-we 3798  df-xp 4133

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