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

Theorem frmin 29562
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 29525 and tz7.5 4888. (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 4835 . . . 4  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
2 sess2 4837 . . . 4  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2anim12d 561 . . 3  |-  ( B 
C_  A  ->  (
( R  Fr  A  /\  R Se  A )  ->  ( R  Fr  B  /\  R Se  B )
) )
4 n0 3793 . . . 4  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
5 predeq3 29488 . . . . . . . . . . 11  |-  ( y  =  b  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B , 
b ) )
65eqeq1d 2456 . . . . . . . . . 10  |-  ( y  =  b  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  B ,  b )  =  (/) ) )
76rspcev 3207 . . . . . . . . 9  |-  ( ( b  e.  B  /\  Pred ( R ,  B ,  b )  =  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
87ex 432 . . . . . . . 8  |-  ( b  e.  B  ->  ( Pred ( R ,  B ,  b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
98adantl 464 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
10 setlikespec 29507 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  Pred ( R ,  B , 
b )  e.  _V )
11 trpredpred 29551 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b ) )
12 ssn0 3817 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b )  /\  Pred ( R ,  B ,  b )  =/=  (/) )  ->  TrPred ( R ,  B ,  b )  =/=  (/) )
1312ex 432 . . . . . . . . . . . . 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 29552 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  TrPred ( R ,  B ,  b )  C_  B )
1614, 15jctild 541 . . . . . . . . . . 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 463 . . . . . . . . 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 29560 . . . . . . . . . . 11  |-  TrPred ( R ,  B ,  b )  e.  _V
20 sseq1 3510 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  C_  B  <->  TrPred ( R ,  B , 
b )  C_  B
) )
21 neeq1 2735 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  =/=  (/)  <->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
2220, 21anbi12d 708 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( c  C_  B  /\  c  =/=  (/) )  <->  ( TrPred ( R ,  B , 
b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
23 predeq2 29487 . . . . . . . . . . . . . . 15  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  Pred ( R , 
c ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
2423eqeq1d 2456 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( Pred ( R ,  c ,  y )  =  (/)  <->  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2524rexeqbi1dv 3060 . . . . . . . . . . . . 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 318 . . . . . . . . . . . 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 314 . . . . . . . . . . 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 29502 . . . . . . . . . . . 12  |-  ( R  Fr  B  <->  A. c
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
29 sp 1864 . . . . . . . . . . . 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 195 . . . . . . . . . . 11  |-  ( R  Fr  B  ->  (
( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
3119, 27, 30vtocl 3158 . . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  TrPred ( R ,  B ,  b )  C_  B )
34 trpredtr 29553 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  B  /\  R Se  B )  ->  (
y  e.  TrPred ( R ,  B ,  b )  ->  Pred ( R ,  B ,  y )  C_  TrPred ( R ,  B ,  b ) ) )
3534imp 427 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  C_  TrPred ( R ,  B ,  b ) )
36 sspred 29492 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 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 2456 . . . . . . . . . . . . 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 223 . . . . . . . . . . . 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 2929 . . . . . . . . . . 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 3551 . . . . . . . . . . 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 56 . . . . . . . . . 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 656 . . . . . . . . 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 44 . . . . . . . 8  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  R  Fr  B
)  ->  ( Pred ( R ,  B , 
b )  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4544an31s 804 . . . . . . 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 2771 . . . . . 6  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
4746ex 432 . . . . 5  |-  ( ( R  Fr  B  /\  R Se  B )  ->  (
b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4847exlimdv 1729 . . . 4  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( E. b  b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
494, 48syl5bi 217 . . 3  |-  ( ( R  Fr  B  /\  R Se  B )  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
503, 49syl6com 35 . 2  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( B  C_  A  ->  ( B  =/=  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) ) )
5150imp32 431 1  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805   _Vcvv 3106    C_ wss 3461   (/)c0 3783    Fr wfr 4824   Se wse 4825   Predcpred 29483   TrPredctrpred 29540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-pred 29484  df-trpred 29541
This theorem is referenced by:  frind  29563
  Copyright terms: Public domain W3C validator