Theorem predpoirr 5708

Description: Given a partial ordering, X is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Assertion

Ref	Expression
predpoirr	|- ( R Po A -> -. X e. Pred ( R , A , X ) )

Proof of Theorem predpoirr

Step	Hyp	Ref	Expression
1		poirr 5046	. . . . 5 |- ( ( R Po A /\ X e. A ) -> -. X R X )
2		elpredg 5694	. . . . . . 7 |- ( ( X e. A /\ X e. A ) -> ( X e. Pred ( R , A , X ) <-> X R X ) )
3	2	anidms 677	. . . . . 6 |- ( X e. A -> ( X e. Pred ( R , A , X ) <-> X R X ) )
4	3	notbid 308	. . . . 5 |- ( X e. A -> ( -. X e. Pred ( R , A , X ) <-> -. X R X ) )
5	1, 4	syl5ibr 236	. . . 4 |- ( X e. A -> ( ( R Po A /\ X e. A ) -> -. X e. Pred ( R , A , X ) ) )
6	5	expd 452	. . 3 |- ( X e. A -> ( R Po A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) ) )
7	6	pm2.43b 55	. 2 |- ( R Po A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) )
8		predel 5697	. . 3 |- ( X e. Pred ( R , A , X ) -> X e. A )
9	8	con3i 150	. 2 |- ( -. X e. A -> -. X e. Pred ( R , A , X ) )
10	7, 9	pm2.61d1 171	1 |- ( R Po A -> -. X e. Pred ( R , A , X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   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: (None)