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Theorem tz6.26 23373
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 4283 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E! y  e.  B  A. x  e.  B  -.  x R y )
2 reurex 2890 . . 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 3383 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  A. x  e.  B  -.  x R y )
5 dfrab3 3351 . . . . . 6  |-  { x  e.  B  |  x R y }  =  ( B  i^i  { x  |  x R y } )
6 vex 2730 . . . . . . 7  |-  y  e. 
_V
76dfpred2 23343 . . . . . 6  |-  Pred ( R ,  B , 
y )  =  ( B  i^i  { x  |  x R y } )
85, 7eqtr4i 2276 . . . . 5  |-  { x  e.  B  |  x R y }  =  Pred ( R ,  B ,  y )
98eqeq1i 2260 . . . 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 2532 . 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 2239    =/= wne 2412   A.wral 2509   E.wrex 2510   E!wreu 2511   {crab 2512    i^i cin 3077    C_ wss 3078   (/)c0 3362   class class class wbr 3920   Se wse 4243    We wwe 4244   Predcpred 23335
This theorem is referenced by:  tz6.26i  23374  wfi  23375
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-pred 23336
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