Theorem poss 2129

Description: Subset theorem for the partial ordering predicate.

Assertion

Ref	Expression
poss	|- (A (_ B -> (R Po B -> R Po A))

Proof of Theorem poss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . . 10 |- (A (_ B -> (x e. A -> x e. B))
2		ssel 1502	. . . . . . . . . 10 |- (A (_ B -> (y e. A -> y e. B))
3	1, 2	anim12d 431	. . . . . . . . 9 |- (A (_ B -> ((x e. A /\ y e. A) -> (x e. B /\ y e. B)))
4		ssel 1502	. . . . . . . . 9 |- (A (_ B -> (z e. A -> z e. B))
5	3, 4	anim12d 431	. . . . . . . 8 |- (A (_ B -> (((x e. A /\ y e. A) /\ z e. A) -> ((x e. B /\ y e. B) /\ z e. B)))
6		df-3an 583	. . . . . . . 8 |- ((x e. A /\ y e. A /\ z e. A) <-> ((x e. A /\ y e. A) /\ z e. A))
7		df-3an 583	. . . . . . . 8 |- ((x e. B /\ y e. B /\ z e. B) <-> ((x e. B /\ y e. B) /\ z e. B))
8	5, 6, 7	3imtr4g 426	. . . . . . 7 |- (A (_ B -> ((x e. A /\ y e. A /\ z e. A) -> (x e. B /\ y e. B /\ z e. B)))
9	8	syl4d 28	. . . . . 6 |- (A (_ B -> (((x e. B /\ y e. B /\ z e. B) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) -> ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
10	9	19.20dv 946	. . . . 5 |- (A (_ B -> (A.z((x e. B /\ y e. B /\ z e. B) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) -> A.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
11	10	19.20dv 946	. . . 4 |- (A (_ B -> (A.yA.z((x e. B /\ y e. B /\ z e. B) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) -> A.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
12	11	19.20dv 946	. . 3 |- (A (_ B -> (A.xA.yA.z((x e. B /\ y e. B /\ z e. B) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))) -> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
13		r3al 1240	. . 3 |- (A.x e. B A.y e. B A.z e. B (-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((x e. B /\ y e. B /\ z e. B) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
14		r3al 1240	. . 3 |- (A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
15	12, 13, 14	3imtr4g 426	. 2 |- (A (_ B -> (A.x e. B A.y e. B A.z e. B (-. xRx /\ ((xRy /\ yRz) -> xRz)) -> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz))))
16		df-po 2128	. 2 |- (R Po B <-> A.x e. B A.y e. B A.z e. B (-. xRx /\ ((xRy /\ yRz) -> xRz)))
17		df-po 2128	. 2 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
18	15, 16, 17	3imtr4g 426	1 |- (A (_ B -> (R Po B -> R Po A))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   /\ w3a 581  A.wal 672   e. wcel 1092  A.wral 1201   (_ wss 1487   class class class wbr 2054   Po wpo 2058

This theorem is referenced by:  poeq2 2131  soss 2140  zornlem6 3608

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-in 1491  df-ss 1492  df-po 2128

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