Theorem tz6.26 5711

Description: All nonempty (possibly proper) subclasses of A, which has a well-founded relation R, have R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
tz6.26	|- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )

Distinct variable groups:   y, A   y, B   y, R

Proof of Theorem tz6.26

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wereu2 5111	. . 3 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E! y e. B A. x e. B -. x R y )
2		reurex 3160	. . 3 |- ( E! y e. B A. x e. B -. x R y -> E. y e. B A. x e. B -. x R y )
3	1, 2	syl 17	. 2 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B A. x e. B -. x R y )
4		rabeq0 3957	. . . 4 |- ( { x e. B | x R y } = (/) <-> A. x e. B -. x R y )
5		dfrab3 3902	. . . . . 6 |- { x e. B | x R y } = ( B i^i { x | x R y } )
6		vex 3203	. . . . . . 7 |- y e. _V
7	6	dfpred2 5689	. . . . . 6 |- Pred ( R , B , y ) = ( B i^i { x | x R y } )
8	5, 7	eqtr4i 2647	. . . . 5 |- { x e. B | x R y } = Pred ( R , B , y )
9	8	eqeq1i 2627	. . . 4 |- ( { x e. B | x R y } = (/) <-> Pred ( R , B , y ) = (/) )
10	4, 9	bitr3i 266	. . 3 |- ( A. x e. B -. x R y <-> Pred ( R , B , y ) = (/) )
11	10	rexbii 3041	. 2 |- ( E. y e. B A. x e. B -. x R y <-> E. y e. B Pred ( R , B , y ) = (/) )
12	3, 11	sylib 208	1 |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Se wse 5071   We wwe 5072   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-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-so 5036  df-fr 5073  df-se 5074  df-we 5075  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:  tz6.26i  5712  wfi  5713  wzel  31771  wzelOLD  31772  wsuclem  31773  wsuclemOLD  31774