Theorem pocnv 31653

Description: The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
pocnv	|- ( R Po A -> `' R Po A )

Proof of Theorem pocnv

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

Step	Hyp	Ref	Expression
1		poirr 5046	. . 3 |- ( ( R Po A /\ x e. A ) -> -. x R x )
2		vex 3203	. . . 4 |- x e. _V
3	2, 2	brcnv 5305	. . 3 |- ( x `' R x <-> x R x )
4	1, 3	sylnibr 319	. 2 |- ( ( R Po A /\ x e. A ) -> -. x `' R x )
5		3anrev 1049	. . . 4 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( z e. A /\ y e. A /\ x e. A ) )
6		potr 5047	. . . 4 |- ( ( R Po A /\ ( z e. A /\ y e. A /\ x e. A ) ) -> ( ( z R y /\ y R x ) -> z R x ) )
7	5, 6	sylan2b 492	. . 3 |- ( ( R Po A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( z R y /\ y R x ) -> z R x ) )
8		vex 3203	. . . . 5 |- y e. _V
9	2, 8	brcnv 5305	. . . 4 |- ( x `' R y <-> y R x )
10		vex 3203	. . . . 5 |- z e. _V
11	8, 10	brcnv 5305	. . . 4 |- ( y `' R z <-> z R y )
12	9, 11	anbi12ci 734	. . 3 |- ( ( x `' R y /\ y `' R z ) <-> ( z R y /\ y R x ) )
13	2, 10	brcnv 5305	. . 3 |- ( x `' R z <-> z R x )
14	7, 12, 13	3imtr4g 285	. 2 |- ( ( R Po A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x `' R y /\ y `' R z ) -> x `' R z ) )
15	4, 14	ispod 5043	1 |- ( R Po A -> `' R Po A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   Po wpo 5033   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-cnv 5122

This theorem is referenced by: (None)