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Theorem frmin 24242
Description: Every (possibly proper) subclass of a class  A with a founded, set-like relation  R has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 24205 and tz7.5 4413. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
frmin  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Distinct variable groups:    y, B    y, R
Allowed substitution hint:    A( y)

Proof of Theorem frmin
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frss 4360 . . . 4  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
2 sess2 4362 . . . 4  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2anim12d 546 . . 3  |-  ( B 
C_  A  ->  (
( R  Fr  A  /\  R Se  A )  ->  ( R  Fr  B  /\  R Se  B )
) )
4 n0 3464 . . . 4  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
5 predeq3 24171 . . . . . . . . . . 11  |-  ( y  =  b  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B , 
b ) )
65eqeq1d 2291 . . . . . . . . . 10  |-  ( y  =  b  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  B ,  b )  =  (/) ) )
76rspcev 2884 . . . . . . . . 9  |-  ( ( b  e.  B  /\  Pred ( R ,  B ,  b )  =  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
87ex 423 . . . . . . . 8  |-  ( b  e.  B  ->  ( Pred ( R ,  B ,  b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
98adantl 452 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
10 setlikespec 24187 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  Pred ( R ,  B , 
b )  e.  _V )
11 trpredpred 24231 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b ) )
12 ssn0 3487 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b )  /\  Pred ( R ,  B ,  b )  =/=  (/) )  ->  TrPred ( R ,  B ,  b )  =/=  (/) )
1312ex 423 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  C_  TrPred ( R ,  B , 
b )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
1411, 13syl 15 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
15 trpredss 24232 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  TrPred ( R ,  B ,  b )  C_  B )
1614, 15jctild 527 . . . . . . . . . . 11  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
1710, 16syl 15 . . . . . . . . . 10  |-  ( ( b  e.  B  /\  R Se  B )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B , 
b )  =/=  (/) ) ) )
1817adantr 451 . . . . . . . . 9  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  R  Fr  B
)  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
19 trpredex 24240 . . . . . . . . . . 11  |-  TrPred ( R ,  B ,  b )  e.  _V
20 sseq1 3199 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  C_  B  <->  TrPred ( R ,  B , 
b )  C_  B
) )
21 neeq1 2454 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  =/=  (/)  <->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
2220, 21anbi12d 691 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( c  C_  B  /\  c  =/=  (/) )  <->  ( TrPred ( R ,  B , 
b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
23 predeq2 24170 . . . . . . . . . . . . . . 15  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  Pred ( R , 
c ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
2423eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( Pred ( R ,  c ,  y )  =  (/)  <->  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2524rexeqbi1dv 2745 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( E. y  e.  c  Pred ( R , 
c ,  y )  =  (/)  <->  E. y  e.  TrPred  ( R ,  B , 
b ) Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2622, 25imbi12d 311 . . . . . . . . . . . 12  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( ( c 
C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) )  <->  ( ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) )  ->  E. y  e.  TrPred  ( R ,  B ,  b ) Pred ( R ,  TrPred ( R ,  B , 
b ) ,  y )  =  (/) ) ) )
2726imbi2d 307 . . . . . . . . . . 11  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( R  Fr  B  ->  ( ( c 
C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )  <-> 
( R  Fr  B  ->  ( ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B , 
b )  =/=  (/) )  ->  E. y  e.  TrPred  ( R ,  B ,  b ) Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) ) ) )
28 dffr4 24182 . . . . . . . . . . . 12  |-  ( R  Fr  B  <->  A. c
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
29 sp 1716 . . . . . . . . . . . 12  |-  ( A. c ( ( c 
C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) )  -> 
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
3028, 29sylbi 187 . . . . . . . . . . 11  |-  ( R  Fr  B  ->  (
( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
3119, 27, 30vtocl 2838 . . . . . . . . . 10  |-  ( R  Fr  B  ->  (
( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) )  ->  E. y  e.  TrPred  ( R ,  B ,  b ) Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
3210, 15syl 15 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  TrPred ( R ,  B ,  b )  C_  B )
3332adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  TrPred ( R ,  B ,  b )  C_  B )
34 trpredtr 24233 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  B  /\  R Se  B )  ->  (
y  e.  TrPred ( R ,  B ,  b )  ->  Pred ( R ,  B ,  y )  C_  TrPred ( R ,  B ,  b ) ) )
3534imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  C_  TrPred ( R ,  B ,  b ) )
36 sspred 24174 . . . . . . . . . . . . . . 15  |-  ( (
TrPred ( R ,  B ,  b )  C_  B  /\  Pred ( R ,  B ,  y )  C_ 
TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
3733, 35, 36syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
3837eqeq1d 2291 . . . . . . . . . . . . 13  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
3938biimprd 214 . . . . . . . . . . . 12  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  ( Pred ( R ,  TrPred ( R ,  B , 
b ) ,  y )  =  (/)  ->  Pred ( R ,  B , 
y )  =  (/) ) )
4039reximdva 2655 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  ( E. y  e.  TrPred  ( R ,  B ,  b ) Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/)  ->  E. y  e.  TrPred  ( R ,  B , 
b ) Pred ( R ,  B , 
y )  =  (/) ) )
41 ssrexv 3238 . . . . . . . . . . 11  |-  ( TrPred ( R ,  B , 
b )  C_  B  ->  ( E. y  e. 
TrPred  ( R ,  B ,  b ) Pred ( R ,  B ,  y )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4232, 40, 41sylsyld 52 . . . . . . . . . 10  |-  ( ( b  e.  B  /\  R Se  B )  ->  ( E. y  e.  TrPred  ( R ,  B ,  b ) Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4331, 42sylan9r 639 . . . . . . . . 9  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  R  Fr  B
)  ->  ( ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4418, 43syld 40 . . . . . . . 8  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  R  Fr  B
)  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4544an31s 781 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
469, 45pm2.61dne 2523 . . . . . 6  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
4746ex 423 . . . . 5  |-  ( ( R  Fr  B  /\  R Se  B )  ->  (
b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4847exlimdv 1664 . . . 4  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( E. b  b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
494, 48syl5bi 208 . . 3  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
503, 49syl6com 31 . 2  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) ) )
5150imp32 422 1  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455    Fr wfr 4349   Se wse 4350   Predcpred 24167   TrPredctrpred 24220
This theorem is referenced by:  frind  24243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-pred 24168  df-trpred 24221
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