Theorem weinxp 2467

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 1502	. . . . . . . . . . . . . 14 |- (z (_ A -> (x e. z -> x e. A))
2		ssel 1502	. . . . . . . . . . . . . 14 |- (z (_ A -> (y e. z -> y e. A))
3	1, 2	anim12d 431	. . . . . . . . . . . . 13 |- (z (_ A -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
4	3	adantr 306	. . . . . . . . . . . 12 |- ((z (_ A /\ -. z = (/)) -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
5		brinxp 2466	. . . . . . . . . . . . 13 |- ((y e. A /\ x e. A) -> (yRx <-> y(R i^i (A X. A))x))
6	5	ancoms 334	. . . . . . . . . . . 12 |- ((x e. A /\ y e. A) -> (yRx <-> y(R i^i (A X. A))x))
7	4, 6	syl6 23	. . . . . . . . . . 11 |- ((z (_ A /\ -. z = (/)) -> ((x e. z /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x)))
8	7	exp3a 292	. . . . . . . . . 10 |- ((z (_ A /\ -. z = (/)) -> (x e. z -> (y e. z -> (yRx <-> y(R i^i (A X. A))x))))
9	8	imp31 280	. . . . . . . . 9 |- ((((z (_ A /\ -. z = (/)) /\ x e. z) /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x))
10	9	negbid 463	. . . . . . . 8 |- ((((z (_ A /\ -. z = (/)) /\ x e. z) /\ y e. z) -> (-. yRx <-> -. y(R i^i (A X. A))x))
11	10	biraldva 1215	. . . . . . 7 |- (((z (_ A /\ -. z = (/)) /\ x e. z) -> (A.y e. z -. yRx <-> A.y e. z -. y(R i^i (A X. A))x))
12	11	birexdva 1216	. . . . . 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 443	. . . . 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	bial 695	. . . 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 2169	. . . 4 |- (R Fr A <-> A.z((z (_ A /\ -. z = (/)) -> E.x e. z A.y e. z -. yRx))
16		df-fr 2169	. . . 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 158	. . 3 |- (R Fr A <-> (R i^i (A X. A)) Fr A)
18		brinxp 2466	. . . . . . 7 |- ((x e. A /\ y e. A) -> (xRy <-> x(R i^i (A X. A))y))
19		pm4.2i 149	. . . . . . 7 |- ((x e. A /\ y e. A) -> (x = y <-> x = y))
20	18, 19, 6	bi3ord 635	. . . . . 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 443	. . . . 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	bi2al 696	. . . 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 1231	. . . 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 1231	. . . 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 158	. . 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 369	. 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 2187	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
28		dfwe2 2187	. 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 158	1 |- (R We A <-> (R i^i (A X. A)) We A)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   \/ w3o 580  A.wal 672   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707   class class class wbr 2054   Fr wfr 2061   We wwe 2062   X. cxp 2408

This theorem is referenced by:  weth 3602

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-xp 2424

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