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Theorem tz6.26 25480
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.
StepHypRef Expression
1 wereu2 4579 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E! y  e.  B  A. x  e.  B  -.  x R y )
2 reurex 2922 . . 3  |-  ( E! y  e.  B  A. x  e.  B  -.  x R y  ->  E. y  e.  B  A. x  e.  B  -.  x R y )
31, 2syl 16 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  A. x  e.  B  -.  x R y )
4 rabeq0 3649 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  A. x  e.  B  -.  x R y )
5 dfrab3 3617 . . . . . 6  |-  { x  e.  B  |  x R y }  =  ( B  i^i  { x  |  x R y } )
6 vex 2959 . . . . . . 7  |-  y  e. 
_V
76dfpred2 25448 . . . . . 6  |-  Pred ( R ,  B , 
y )  =  ( B  i^i  { x  |  x R y } )
85, 7eqtr4i 2459 . . . . 5  |-  { x  e.  B  |  x R y }  =  Pred ( R ,  B ,  y )
98eqeq1i 2443 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  Pred ( R ,  B ,  y )  =  (/) )
104, 9bitr3i 243 . . 3  |-  ( A. x  e.  B  -.  x R y  <->  Pred ( R ,  B ,  y )  =  (/) )
1110rexbii 2730 . 2  |-  ( E. y  e.  B  A. x  e.  B  -.  x R y  <->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
123, 11sylib 189 1  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709    i^i cin 3319    C_ wss 3320   (/)c0 3628   class class class wbr 4212   Se wse 4539    We wwe 4540   Predcpred 25438
This theorem is referenced by:  tz6.26i  25481  wfi  25482  wzel  25575  wsuclem  25576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-pred 25439
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