Theorem wepo 4360

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 4359	. . . 4 |- ( R We A -> R Fr A )
2		frirrg 4351	. . . 4 |- ( ( R Fr A /\ A e. V /\ x e. A ) -> -. x R x )
3	1, 2	syl3an1 1271	. . 3 |- ( ( R We A /\ A e. V /\ x e. A ) -> -. x R x )
4	3	3expa 1203	. 2 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> -. x R x )
5		df-3an 980	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
6		df-wetr 4335	. . . . . . . . . 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 275	. . . . . . . . 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 276	. . . . . . . 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 2565	. . . . . . 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 2565	. . . . . 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 399	. . . . 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 2565	. . . 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 399	. . 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 287	. 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 4305	1 |- ( ( R We A /\ A e. V ) -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   A.wral 2455   class class class wbr 4004   Po wpo 4295   Fr wfr 4329   We wwe 4331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4122

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-po 4297  df-frfor 4332  df-frind 4333  df-wetr 4335

This theorem is referenced by: (None)