Theorem poss 4026

Description: Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Assertion

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

Proof of Theorem poss

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 2998	. . 3 |- (A C_ B -> (A.x e. B A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.x e. A A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
2		ssralv 2998	. . . . 5 |- (A C_ B -> (A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.y e. A A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
3		ssralv 2998	. . . . . 6 |- (A C_ B -> (A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
4	3	ralimdv 2382	. . . . 5 |- (A C_ B -> (A.y e. A A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.y e. A A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
5	2, 4	syld 40	. . . 4 |- (A C_ B -> (A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.y e. A A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
6	5	ralimdv 2382	. . 3 |- (A C_ B -> (A.x e. A A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.x e. A A.y e. A A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
7	1, 6	syld 40	. 2 |- (A C_ B -> (A.x e. B A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)) -> A.x e. A A.y e. A A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz))))
8		df-po 4024	. 2 |- ( R Po B <-> A.x e. B A.y e. B A.z e. B (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)))
9		df-po 4024	. 2 |- ( R Po A <-> A.x e. A A.y e. A A.z e. A (-. x Rx /\ ((x Ry /\ y Rz) -> x Rz)))
10	7, 8, 9	3imtr4g 194	1 |- (A C_ B -> ( R Po B -> R Po A))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97  A.wral 2300   C_ wss 2911   class class class wbr 3755   Po wpo 4022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024

This theorem is referenced by:  poeq2  4028  soss  4042