Theorem wepo 4276

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 4275	. . . 4 |- ( R We A -> R Fr A )
2		frirrg 4267	. . . 4 |- ( ( R Fr A /\ A e. V /\ x e. A ) -> -. x R x )
3	1, 2	syl3an1 1249	. . 3 |- ( ( R We A /\ A e. V /\ x e. A ) -> -. x R x )
4	3	3expa 1181	. 2 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> -. x R x )
5		df-3an 964	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
6		df-wetr 4251	. . . . . . . . . 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 2518	. . . . . . 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 2518	. . . . . 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 396	. . . . 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 2518	. . . 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 396	. . 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 4221	1 |- ( ( R We A /\ A e. V ) -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 962   e. wcel 1480   A.wral 2414   class class class wbr 3924   Po wpo 4211   Fr wfr 4245   We wwe 4247

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-po 4213  df-frfor 4248  df-frind 4249  df-wetr 4251

This theorem is referenced by: (None)