Theorem predfrirr 5416

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 4816	. . . . 5 |- ( ( R Fr A /\ X e. A ) -> -. X R X )
2		elpredg 5401	. . . . . . 7 |- ( ( X e. A /\ X e. A ) -> ( X e. Pred ( R , A , X ) <-> X R X ) )
3	2	anidms 657	. . . . . 6 |- ( X e. A -> ( X e. Pred ( R , A , X ) <-> X R X ) )
4	3	notbid 301	. . . . 5 |- ( X e. A -> ( -. X e. Pred ( R , A , X ) <-> -. X R X ) )
5	1, 4	syl5ibr 229	. . . 4 |- ( X e. A -> ( ( R Fr A /\ X e. A ) -> -. X e. Pred ( R , A , X ) ) )
6	5	expd 443	. . 3 |- ( X e. A -> ( R Fr A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) ) )
7	6	pm2.43b 51	. 2 |- ( R Fr A -> ( X e. A -> -. X e. Pred ( R , A , X ) ) )
8		predel 5404	. . 3 |- ( X e. Pred ( R , A , X ) -> X e. A )
9	8	con3i 142	. 2 |- ( -. X e. A -> -. X e. Pred ( R , A , X ) )
10	7, 9	pm2.61d1 164	1 |- ( R Fr A -> -. X e. Pred ( R , A , X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   e. wcel 1904   class class class wbr 4395   Fr wfr 4795   Predcpred 5386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-fr 4798  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387

This theorem is referenced by:  wfrlem14  7067