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Theorem frmin 23644
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 23607 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
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
Allowed substitution hint:    A( y)

Proof of Theorem frmin
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 548 . . 3  |-  ( B 
C_  A  ->  (
( R  Fr  A  /\  R Se  A )  ->  ( R  Fr  B  /\  R Se  B )
) )
4 n0 3466 . . . 4  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
5 predeq3 23573 . . . . . . . . . . 11  |-  ( y  =  b  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B , 
b ) )
65eqeq1d 2293 . . . . . . . . . 10  |-  ( y  =  b  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  B ,  b )  =  (/) ) )
76rspcev 2886 . . . . . . . . 9  |-  ( ( b  e.  B  /\  Pred ( R ,  B ,  b )  =  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
87ex 425 . . . . . . . 8  |-  ( b  e.  B  ->  ( Pred ( R ,  B ,  b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
98adantl 454 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
10 setlikespec 23589 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  Pred ( R ,  B , 
b )  e.  _V )
11 trpredpred 23633 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b ) )
12 ssn0 3489 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b )  /\  Pred ( R ,  B ,  b )  =/=  (/) )  ->  TrPred ( R ,  B ,  b )  =/=  (/) )
1312ex 425 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  C_  TrPred ( R ,  B , 
b )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
1411, 13syl 17 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
15 trpredss 23634 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  TrPred ( R ,  B ,  b )  C_  B )
1614, 15jctild 529 . . . . . . . . . . 11  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
1710, 16syl 17 . . . . . . . . . 10  |-  ( ( b  e.  B  /\  R Se  B )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B , 
b )  =/=  (/) ) ) )
1817adantr 453 . . . . . . . . 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 23642 . . . . . . . . . . 11  |-  TrPred ( R ,  B ,  b )  e.  _V
20 sseq1 3201 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  C_  B  <->  TrPred ( R ,  B , 
b )  C_  B
) )
21 neeq1 2456 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  =/=  (/)  <->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
2220, 21anbi12d 693 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( c  C_  B  /\  c  =/=  (/) )  <->  ( TrPred ( R ,  B , 
b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
23 predeq2 23572 . . . . . . . . . . . . . . 15  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  Pred ( R , 
c ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
2423eqeq1d 2293 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( Pred ( R ,  c ,  y )  =  (/)  <->  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2524rexeqbi1dv 2747 . . . . . . . . . . . . 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 313 . . . . . . . . . . . 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 309 . . . . . . . . . . 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 23584 . . . . . . . . . . . 12  |-  ( R  Fr  B  <->  A. c
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
29 sp 1717 . . . . . . . . . . . 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 189 . . . . . . . . . . 11  |-  ( R  Fr  B  ->  (
( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
3119, 27, 30vtocl 2840 . . . . . . . . . 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 17 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  TrPred ( R ,  B ,  b )  C_  B )
3332adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  TrPred ( R ,  B ,  b )  C_  B )
34 trpredtr 23635 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  B  /\  R Se  B )  ->  (
y  e.  TrPred ( R ,  B ,  b )  ->  Pred ( R ,  B ,  y )  C_  TrPred ( R ,  B ,  b ) ) )
3534imp 420 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  C_  TrPred ( R ,  B ,  b ) )
36 sspred 23576 . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . 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 2293 . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 2657 . . . . . . . . . . 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 3240 . . . . . . . . . . 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 54 . . . . . . . . . 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 641 . . . . . . . . 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 42 . . . . . . . 8  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  R  Fr  B
)  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4544an31s 783 . . . . . . 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 2525 . . . . . 6  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
4746ex 425 . . . . 5  |-  ( ( R  Fr  B  /\  R Se  B )  ->  (
b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4847exlimdv 1665 . . . 4  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( E. b  b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
494, 48syl5bi 210 . . 3  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
503, 49syl6com 33 . 2  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) ) )
5150imp32 424 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 6    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685    =/= wne 2448   E.wrex 2546   _Vcvv 2790    C_ wss 3154   (/)c0 3457    Fr wfr 4349   Se wse 4350   Predcpred 23569   TrPredctrpred 23622
This theorem is referenced by:  frind  23645
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384  df-rdg 6419  df-pred 23570  df-trpred 23623
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