Theorem wepo 4153

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 4152	. . . 4 |- ( R We A -> R Fr A )
2		frirrg 4144	. . . 4 |- ( ( R Fr A /\ A e. V /\ x e. A ) -> -. x R x )
3	1, 2	syl3an1 1205	. . 3 |- ( ( R We A /\ A e. V /\ x e. A ) -> -. x R x )
4	3	3expa 1141	. 2 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> -. x R x )
5		df-3an 924	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
6		df-wetr 4128	. . . . . . . . . 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 269	. . . . . . . . 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 270	. . . . . . . 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 2457	. . . . . . 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 2457	. . . . . 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 391	. . . . 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 2457	. . . 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 391	. . 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 281	. 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 4098	1 |- ( ( R We A /\ A e. V ) -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   /\ w3a 922   e. wcel 1436   A.wral 2355   class class class wbr 3814   Po wpo 4088   Fr wfr 4122   We wwe 4124

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-po 4090  df-frfor 4125  df-frind 4126  df-wetr 4128

This theorem is referenced by: (None)