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Theorem tz6.26 23539
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
StepHypRef Expression
1 wereu2 4327 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E! y  e.  B  A. x  e.  B  -.  x R y )
2 reurex 2904 . . 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 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 3418 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  A. x  e.  B  -.  x R y )
5 dfrab3 3386 . . . . . 6  |-  { x  e.  B  |  x R y }  =  ( B  i^i  { x  |  x R y } )
6 vex 2743 . . . . . . 7  |-  y  e. 
_V
76dfpred2 23509 . . . . . 6  |-  Pred ( R ,  B , 
y )  =  ( B  i^i  { x  |  x R y } )
85, 7eqtr4i 2279 . . . . 5  |-  { x  e.  B  |  x R y }  =  Pred ( R ,  B ,  y )
98eqeq1i 2263 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  Pred ( R ,  B ,  y )  =  (/) )
104, 9bitr3i 244 . . 3  |-  ( A. x  e.  B  -.  x R y  <->  Pred ( R ,  B ,  y )  =  (/) )
1110rexbii 2539 . 2  |-  ( E. y  e.  B  A. x  e.  B  -.  x R y  <->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
123, 11sylib 190 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 5    -> wi 6    /\ wa 360    = wceq 1619   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517   E!wreu 2518   {crab 2519    i^i cin 3093    C_ wss 3094   (/)c0 3397   class class class wbr 3963   Se wse 4287    We wwe 4288   Predcpred 23501
This theorem is referenced by:  tz6.26i  23540  wfi  23541
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-pred 23502
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