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Theorem tz6.26 24207
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 4392 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E! y  e.  B  A. x  e.  B  -.  x R y )
2 reurex 2756 . . 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 15 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  A. x  e.  B  -.  x R y )
4 rabeq0 3478 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  A. x  e.  B  -.  x R y )
5 dfrab3 3446 . . . . . 6  |-  { x  e.  B  |  x R y }  =  ( B  i^i  { x  |  x R y } )
6 vex 2793 . . . . . . 7  |-  y  e. 
_V
76dfpred2 24177 . . . . . 6  |-  Pred ( R ,  B , 
y )  =  ( B  i^i  { x  |  x R y } )
85, 7eqtr4i 2308 . . . . 5  |-  { x  e.  B  |  x R y }  =  Pred ( R ,  B ,  y )
98eqeq1i 2292 . . . 4  |-  ( { x  e.  B  |  x R y }  =  (/)  <->  Pred ( R ,  B ,  y )  =  (/) )
104, 9bitr3i 242 . . 3  |-  ( A. x  e.  B  -.  x R y  <->  Pred ( R ,  B ,  y )  =  (/) )
1110rexbii 2570 . 2  |-  ( E. y  e.  B  A. x  e.  B  -.  x R y  <->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
123, 11sylib 188 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 358    = wceq 1625   {cab 2271    =/= wne 2448   A.wral 2545   E.wrex 2546   E!wreu 2547   {crab 2549    i^i cin 3153    C_ wss 3154   (/)c0 3457   class class class wbr 4025   Se wse 4352    We wwe 4353   Predcpred 24169
This theorem is referenced by:  tz6.26i  24208  wfi  24209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-xp 4697  df-cnv 4699  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-pred 24170
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