Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frmin Structured version   Unicode version

Theorem frmin 25517
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 25480 and tz7.5 4602. (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 4549 . . . 4  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
2 sess2 4551 . . . 4  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2anim12d 547 . . 3  |-  ( B 
C_  A  ->  (
( R  Fr  A  /\  R Se  A )  ->  ( R  Fr  B  /\  R Se  B )
) )
4 n0 3637 . . . 4  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
5 predeq3 25443 . . . . . . . . . . 11  |-  ( y  =  b  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B , 
b ) )
65eqeq1d 2444 . . . . . . . . . 10  |-  ( y  =  b  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  B ,  b )  =  (/) ) )
76rspcev 3052 . . . . . . . . 9  |-  ( ( b  e.  B  /\  Pred ( R ,  B ,  b )  =  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
87ex 424 . . . . . . . 8  |-  ( b  e.  B  ->  ( Pred ( R ,  B ,  b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
98adantl 453 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
10 setlikespec 25462 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  Pred ( R ,  B , 
b )  e.  _V )
11 trpredpred 25506 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b ) )
12 ssn0 3660 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b )  /\  Pred ( R ,  B ,  b )  =/=  (/) )  ->  TrPred ( R ,  B ,  b )  =/=  (/) )
1312ex 424 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  C_  TrPred ( R ,  B , 
b )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
1411, 13syl 16 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
15 trpredss 25507 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  TrPred ( R ,  B ,  b )  C_  B )
1614, 15jctild 528 . . . . . . . . . . 11  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
1710, 16syl 16 . . . . . . . . . 10  |-  ( ( b  e.  B  /\  R Se  B )  ->  ( Pred ( R ,  B ,  b )  =/=  (/)  ->  ( TrPred ( R ,  B ,  b )  C_  B  /\  TrPred ( R ,  B , 
b )  =/=  (/) ) ) )
1817adantr 452 . . . . . . . . 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 25515 . . . . . . . . . . 11  |-  TrPred ( R ,  B ,  b )  e.  _V
20 sseq1 3369 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  C_  B  <->  TrPred ( R ,  B , 
b )  C_  B
) )
21 neeq1 2609 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  =/=  (/)  <->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
2220, 21anbi12d 692 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( c  C_  B  /\  c  =/=  (/) )  <->  ( TrPred ( R ,  B , 
b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
23 predeq2 25442 . . . . . . . . . . . . . . 15  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  Pred ( R , 
c ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
2423eqeq1d 2444 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( Pred ( R ,  c ,  y )  =  (/)  <->  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2524rexeqbi1dv 2913 . . . . . . . . . . . . 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 312 . . . . . . . . . . . 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 308 . . . . . . . . . . 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 25457 . . . . . . . . . . . 12  |-  ( R  Fr  B  <->  A. c
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
29 sp 1763 . . . . . . . . . . . 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 188 . . . . . . . . . . 11  |-  ( R  Fr  B  ->  (
( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
3119, 27, 30vtocl 3006 . . . . . . . . . 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 16 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  TrPred ( R ,  B ,  b )  C_  B )
3332adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  TrPred ( R ,  B ,  b )  C_  B )
34 trpredtr 25508 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  B  /\  R Se  B )  ->  (
y  e.  TrPred ( R ,  B ,  b )  ->  Pred ( R ,  B ,  y )  C_  TrPred ( R ,  B ,  b ) ) )
3534imp 419 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  C_  TrPred ( R ,  B ,  b ) )
36 sspred 25447 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 2818 . . . . . . . . . . 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 3408 . . . . . . . . . . 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 640 . . . . . . . . 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 782 . . . . . . 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 2681 . . . . . 6  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
4746ex 424 . . . . 5  |-  ( ( R  Fr  B  /\  R Se  B )  ->  (
b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4847exlimdv 1646 . . . 4  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( E. b  b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
494, 48syl5bi 209 . . 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 423 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 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628    Fr wfr 4538   Se wse 4539   Predcpred 25438   TrPredctrpred 25495
This theorem is referenced by:  frind  25518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-pred 25439  df-trpred 25496
  Copyright terms: Public domain W3C validator