Theorem wepo 4462

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 4461	. . . 4 |- ( R We A -> R Fr A )
2		frirrg 4453	. . . 4 |- ( ( R Fr A /\ A e. V /\ x e. A ) -> -. x R x )
3	1, 2	syl3an1 1307	. . 3 |- ( ( R We A /\ A e. V /\ x e. A ) -> -. x R x )
4	3	3expa 1230	. 2 |- ( ( ( R We A /\ A e. V ) /\ x e. A ) -> -. x R x )
5		df-3an 1007	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
6		df-wetr 4437	. . . . . . . . . 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 2621	. . . . . . 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 2621	. . . . . 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 2621	. . . 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 4407	1 |- ( ( R We A /\ A e. V ) -> R Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2202   A.wral 2511   class class class wbr 4093   Po wpo 4397   Fr wfr 4431   We wwe 4433

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-po 4399  df-frfor 4434  df-frind 4435  df-wetr 4437

This theorem is referenced by: (None)