Theorem predpo 5698

Description: Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Assertion

Ref	Expression
predpo	|- ( ( R Po A /\ X e. A ) -> ( Y e. Pred ( R , A , X ) -> Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) )

Proof of Theorem predpo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		predel 5697	. 2 |- ( Y e. Pred ( R , A , X ) -> Y e. A )
2		elpredg 5694	. . . . . . . . . . 11 |- ( ( X e. A /\ Y e. A ) -> ( Y e. Pred ( R , A , X ) <-> Y R X ) )
3	2	adantll 750	. . . . . . . . . 10 |- ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> ( Y e. Pred ( R , A , X ) <-> Y R X ) )
4		potr 5047	. . . . . . . . . . . . . . . 16 |- ( ( R Po A /\ ( z e. A /\ Y e. A /\ X e. A ) ) -> ( ( z R Y /\ Y R X ) -> z R X ) )
5	4	3exp2 1285	. . . . . . . . . . . . . . 15 |- ( R Po A -> ( z e. A -> ( Y e. A -> ( X e. A -> ( ( z R Y /\ Y R X ) -> z R X ) ) ) ) )
6	5	com24 95	. . . . . . . . . . . . . 14 |- ( R Po A -> ( X e. A -> ( Y e. A -> ( z e. A -> ( ( z R Y /\ Y R X ) -> z R X ) ) ) ) )
7	6	imp31 448	. . . . . . . . . . . . 13 |- ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> ( z e. A -> ( ( z R Y /\ Y R X ) -> z R X ) ) )
8	7	com13 88	. . . . . . . . . . . 12 |- ( ( z R Y /\ Y R X ) -> ( z e. A -> ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> z R X ) ) )
9	8	ex 450	. . . . . . . . . . 11 |- ( z R Y -> ( Y R X -> ( z e. A -> ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> z R X ) ) ) )
10	9	com14 96	. . . . . . . . . 10 |- ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> ( Y R X -> ( z e. A -> ( z R Y -> z R X ) ) ) )
11	3, 10	sylbid 230	. . . . . . . . 9 |- ( ( ( R Po A /\ X e. A ) /\ Y e. A ) -> ( Y e. Pred ( R , A , X ) -> ( z e. A -> ( z R Y -> z R X ) ) ) )
12	11	ex 450	. . . . . . . 8 |- ( ( R Po A /\ X e. A ) -> ( Y e. A -> ( Y e. Pred ( R , A , X ) -> ( z e. A -> ( z R Y -> z R X ) ) ) ) )
13	12	com23 86	. . . . . . 7 |- ( ( R Po A /\ X e. A ) -> ( Y e. Pred ( R , A , X ) -> ( Y e. A -> ( z e. A -> ( z R Y -> z R X ) ) ) ) )
14	13	3imp 1256	. . . . . 6 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> ( z e. A -> ( z R Y -> z R X ) ) )
15	14	imdistand 728	. . . . 5 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> ( ( z e. A /\ z R Y ) -> ( z e. A /\ z R X ) ) )
16		vex 3203	. . . . . . 7 |- z e. _V
17	16	elpred 5693	. . . . . 6 |- ( Y e. A -> ( z e. Pred ( R , A , Y ) <-> ( z e. A /\ z R Y ) ) )
18	17	3ad2ant3 1084	. . . . 5 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> ( z e. Pred ( R , A , Y ) <-> ( z e. A /\ z R Y ) ) )
19	16	elpred 5693	. . . . . . 7 |- ( X e. A -> ( z e. Pred ( R , A , X ) <-> ( z e. A /\ z R X ) ) )
20	19	adantl 482	. . . . . 6 |- ( ( R Po A /\ X e. A ) -> ( z e. Pred ( R , A , X ) <-> ( z e. A /\ z R X ) ) )
21	20	3ad2ant1 1082	. . . . 5 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> ( z e. Pred ( R , A , X ) <-> ( z e. A /\ z R X ) ) )
22	15, 18, 21	3imtr4d 283	. . . 4 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> ( z e. Pred ( R , A , Y ) -> z e. Pred ( R , A , X ) ) )
23	22	ssrdv 3609	. . 3 |- ( ( ( R Po A /\ X e. A ) /\ Y e. Pred ( R , A , X ) /\ Y e. A ) -> Pred ( R , A , Y ) C_ Pred ( R , A , X ) )
24	23	3exp 1264	. 2 |- ( ( R Po A /\ X e. A ) -> ( Y e. Pred ( R , A , X ) -> ( Y e. A -> Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) ) )
25	1, 24	mpdi 45	1 |- ( ( R Po A /\ X e. A ) -> ( Y e. Pred ( R , A , X ) -> Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   class class class wbr 4653   Po wpo 5033   Predcpred 5679

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-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  predso  5699  trpredpo  31735