Theorem poss 2841

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 2063	. . . . . . . 8 |- (A (_ B -> (x e. A -> x e. B))
2		ssel 2063	. . . . . . . 8 |- (A (_ B -> (y e. A -> y e. B))
3		ssel 2063	. . . . . . . 8 |- (A (_ B -> (z e. A -> z e. B))
4	1, 2, 3	3anim123d 900	. . . . . . 7 |- (A (_ B -> ((x e. A /\ y e. A /\ z e. A) -> (x e. B /\ y e. B /\ z e. B)))
5	4	imim1d 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)))))
6	5	19.20dv 1289	. . . . 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)))))
7	6	19.20dv 1289	. . . 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)))))
8	7	19.20dv 1289	. . 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)))))
9		r3al 1690	. . 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))))
10		r3al 1690	. . 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))))
11	8, 9, 10	3imtr4g 553	. 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))))
12		df-po 2840	. 2 |- (R Po B <-> A.x e. B A.y e. B A.z e. B (-. xRx /\ ((xRy /\ yRz) -> xRz)))
13		df-po 2840	. 2 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
14	11, 12, 13	3imtr4g 553	1 |- (A (_ B -> (R Po B -> R Po A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   e. wcel 958  A.wral 1645   (_ wss 2047   class class class wbr 2619   Po wpo 2838

This theorem is referenced by:  poeq2 2843  soss 2852  zorn2lem6 4793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-in 2051  df-ss 2053  df-po 2840

Copyright terms: Public domain