Theorem wepo 4337

Description: A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)

Assertion

Ref	Expression
wepo	|- ( ( R We A /\ A e. V ) -> R Po A )

Proof of Theorem wepo

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

Step	Hyp	Ref	Expression
1		wefr 4336	. . . 4 |- ( R We A -> R Fr A )
2		frirrg 4328	. . . 4 |- ( ( R Fr A /\ A e. V /\ x e. A ) -> -. x R x )
3	1, 2	syl3an1 1261	. . 3 |- ( ( R We A /\ A e. V /\ x e. A ) -> -. x R x )
4	3	3expa 1193	. 2 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> -. x R x )
5		df-3an 970	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
6		df-wetr 4312	. . . . . . . . . 10 |- ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) )
7	6	simprbi 273	. . . . . . . . 9 |- ( R We A -> A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
8	7	adantr 274	. . . . . . . 8 |- ( ( R We A /\ A e. V ) -> A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
9	8	r19.21bi 2554	. . . . . . 7 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
10	9	r19.21bi 2554	. . . . . 6 |- ( ( ( ( R We A /\ A e. V ) /\ x e. A ) /\ y e. A ) -> A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
11	10	anasss 397	. . . . 5 |- ( ( ( R We A /\ A e. V ) /\ ( x e. A /\ y e. A ) ) -> A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
12	11	r19.21bi 2554	. . . 4 |- ( ( ( ( R We A /\ A e. V ) /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )
13	12	anasss 397	. . 3 |- ( ( ( R We A /\ A e. V ) /\ ( ( x e. A /\ y e. A ) /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
14	5, 13	sylan2b 285	. 2 |- ( ( ( R We A /\ A e. V ) /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x R y /\ y R z ) -> x R z ) )
15	4, 14	ispod 4282	1 |- ( ( R We A /\ A e. V ) -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 968   e. wcel 2136   A.wral 2444   class class class wbr 3982   Po wpo 4272   Fr wfr 4306   We wwe 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-po 4274  df-frfor 4309  df-frind 4310  df-wetr 4312

This theorem is referenced by: (None)