Theorem weinxp 3228

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 2059	. . . . . . . . . . . . . 14 |- (z (_ A -> (x e. z -> x e. A))
2		ssel 2059	. . . . . . . . . . . . . 14 |- (z (_ A -> (y e. z -> y e. A))
3	1, 2	anim12d 557	. . . . . . . . . . . . 13 |- (z (_ A -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
4		brinxp 3227	. . . . . . . . . . . . . 14 |- ((y e. A /\ x e. A) -> (yRx <-> y(R i^i (A X. A))x))
5	4	ancoms 436	. . . . . . . . . . . . 13 |- ((x e. A /\ y e. A) -> (yRx <-> y(R i^i (A X. A))x))
6	3, 5	syl6 22	. . . . . . . . . . . 12 |- (z (_ A -> ((x e. z /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x)))
7	6	exp3a 375	. . . . . . . . . . 11 |- (z (_ A -> (x e. z -> (y e. z -> (yRx <-> y(R i^i (A X. A))x))))
8	7	imp31 362	. . . . . . . . . 10 |- (((z (_ A /\ x e. z) /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x))
9	8	negbid 610	. . . . . . . . 9 |- (((z (_ A /\ x e. z) /\ y e. z) -> (-. yRx <-> -. y(R i^i (A X. A))x))
10	9	ralbidva 1656	. . . . . . . 8 |- ((z (_ A /\ x e. z) -> (A.y e. z -. yRx <-> A.y e. z -. y(R i^i (A X. A))x))
11	10	rexbidva 1657	. . . . . . 7 |- (z (_ 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 389	. . . . . 6 |- ((z (_ 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 583	. . . . 5 |- (((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> ((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
14	13	albii 997	. . . 4 |- (A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
15		df-fr 2912	. . . 4 |- (R Fr A <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx))
16		df-fr 2912	. . . 4 |- ((R i^i (A X. A)) Fr A <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
17	14, 15, 16	3bitr4 183	. . 3 |- (R Fr A <-> (R i^i (A X. A)) Fr A)
18		brinxp 3227	. . . . . . 7 |- ((x e. A /\ y e. A) -> (xRy <-> x(R i^i (A X. A))y))
19		pm4.2i 171	. . . . . . 7 |- ((x e. A /\ y e. A) -> (x = y <-> x = y))
20	18, 19, 5	3orbi123d 890	. . . . . 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 583	. . . . 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 998	. . . 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 1673	. . . 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 1673	. . . 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	3bitr4 183	. . 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 482	. 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 2930	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
28		dfwe2 2930	. 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	3bitr4 183	1 |- (R We A <-> (R i^i (A X. A)) We A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 773  A.wal 952   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643   i^i cin 2042   (_ wss 2043  (/)c0 2276   class class class wbr 2614   Fr wfr 2910   We wwe 2911   X. cxp 3163

This theorem is referenced by:  weth 4767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-xp 3179

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