Theorem predfrirr 29521

Description: Given a well-founded relationship, X is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)

Assertion

Ref	Expression
predfrirr	|- ( R Fr A -> -. X e. Pred ( R , A , X ) )

Proof of Theorem predfrirr

Step	Hyp	Ref	Expression
1		frirr 4845	. . . . 5 |- ( ( R Fr A /\ X e. A ) -> -. X R X )
2		elpredg 29501	. . . . . . 7 |- ( ( X e. A /\ X e. A ) -> ( X e. Pred ( R , A , X ) <-> X R X ) )
3	2	anidms 643	. . . . . 6 |- ( X e. A -> ( X e. Pred ( R , A , X ) <-> X R X ) )
4	3	notbid 292	. . . . 5 |- ( X e. A -> ( -. X e. Pred ( R , A , X ) <-> -. X R X ) )
5	1, 4	syl5ibr 221	. . . 4 |- ( X e. A -> ( ( R Fr A /\ X e. A ) -> -. X e. Pred ( R , A , X ) ) )
6	5	expd 434	. . 3 |- ( X e. A -> ( R Fr A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) ) )
7	6	pm2.43b 50	. 2 |- ( R Fr A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) )
8		predel 29506	. . 3 |- ( X e. Pred ( R , A , X ) -> X e. A )
9	8	con3i 135	. 2 |- ( -. X e. A -> -. X e. Pred ( R , A , X ) )
10	7, 9	pm2.61d1 159	1 |- ( R Fr A -> -. X e. Pred ( R , A , X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   e. wcel 1823   class class class wbr 4439   Fr wfr 4824   Predcpred 29486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-fr 4827  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-pred 29487

This theorem is referenced by:  wfrlem14  29599