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

Theorem frmin 27715
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 27678 and tz7.5 4752. (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 4699 . . . 4  |-  ( B 
C_  A  ->  ( R  Fr  A  ->  R  Fr  B ) )
2 sess2 4701 . . . 4  |-  ( B 
C_  A  ->  ( R Se  A  ->  R Se  B
) )
31, 2anim12d 563 . . 3  |-  ( B 
C_  A  ->  (
( R  Fr  A  /\  R Se  A )  ->  ( R  Fr  B  /\  R Se  B )
) )
4 n0 3658 . . . 4  |-  ( B  =/=  (/)  <->  E. b  b  e.  B )
5 predeq3 27641 . . . . . . . . . . 11  |-  ( y  =  b  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B , 
b ) )
65eqeq1d 2451 . . . . . . . . . 10  |-  ( y  =  b  ->  ( Pred ( R ,  B ,  y )  =  (/) 
<-> 
Pred ( R ,  B ,  b )  =  (/) ) )
76rspcev 3085 . . . . . . . . 9  |-  ( ( b  e.  B  /\  Pred ( R ,  B ,  b )  =  (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
87ex 434 . . . . . . . 8  |-  ( b  e.  B  ->  ( Pred ( R ,  B ,  b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
98adantl 466 . . . . . . 7  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  ( Pred ( R ,  B , 
b )  =  (/)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
10 setlikespec 27660 . . . . . . . . . . 11  |-  ( ( b  e.  B  /\  R Se  B )  ->  Pred ( R ,  B , 
b )  e.  _V )
11 trpredpred 27704 . . . . . . . . . . . . 13  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b ) )
12 ssn0 3682 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R ,  B ,  b )  C_ 
TrPred ( R ,  B ,  b )  /\  Pred ( R ,  B ,  b )  =/=  (/) )  ->  TrPred ( R ,  B ,  b )  =/=  (/) )
1312ex 434 . . . . . . . . . . . . 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 27705 . . . . . . . . . . . 12  |-  ( Pred ( R ,  B ,  b )  e. 
_V  ->  TrPred ( R ,  B ,  b )  C_  B )
1614, 15jctild 543 . . . . . . . . . . 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 465 . . . . . . . . 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 27713 . . . . . . . . . . 11  |-  TrPred ( R ,  B ,  b )  e.  _V
20 sseq1 3389 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  C_  B  <->  TrPred ( R ,  B , 
b )  C_  B
) )
21 neeq1 2628 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( c  =/=  (/)  <->  TrPred ( R ,  B ,  b )  =/=  (/) ) )
2220, 21anbi12d 710 . . . . . . . . . . . . 13  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( ( c  C_  B  /\  c  =/=  (/) )  <->  ( TrPred ( R ,  B , 
b )  C_  B  /\  TrPred ( R ,  B ,  b )  =/=  (/) ) ) )
23 predeq2 27640 . . . . . . . . . . . . . . 15  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  Pred ( R , 
c ,  y )  =  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y ) )
2423eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( c  =  TrPred ( R ,  B ,  b )  ->  ( Pred ( R ,  c ,  y )  =  (/)  <->  Pred ( R ,  TrPred ( R ,  B ,  b ) ,  y )  =  (/) ) )
2524rexeqbi1dv 2938 . . . . . . . . . . . . 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 320 . . . . . . . . . . . 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 316 . . . . . . . . . . 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 27655 . . . . . . . . . . . 12  |-  ( R  Fr  B  <->  A. c
( ( c  C_  B  /\  c  =/=  (/) )  ->  E. y  e.  c  Pred ( R ,  c ,  y )  =  (/) ) )
29 sp 1794 . . . . . . . . . . . 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 3036 . . . . . . . . . 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 465 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  TrPred ( R ,  B ,  b )  C_  B )
34 trpredtr 27706 . . . . . . . . . . . . . . . 16  |-  ( ( b  e.  B  /\  R Se  B )  ->  (
y  e.  TrPred ( R ,  B ,  b )  ->  Pred ( R ,  B ,  y )  C_  TrPred ( R ,  B ,  b ) ) )
3534imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( b  e.  B  /\  R Se  B )  /\  y  e.  TrPred ( R ,  B ,  b ) )  ->  Pred ( R ,  B , 
y )  C_  TrPred ( R ,  B ,  b ) )
36 sspred 27645 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . 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 2840 . . . . . . . . . . 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 3429 . . . . . . . . . . 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 658 . . . . . . . . 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 2700 . . . . . 6  |-  ( ( ( R  Fr  B  /\  R Se  B )  /\  b  e.  B
)  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
4746ex 434 . . . . 5  |-  ( ( R  Fr  B  /\  R Se  B )  ->  (
b  e.  B  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) ) )
4847exlimdv 1690 . . . 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 433 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 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2618   E.wrex 2728   _Vcvv 2984    C_ wss 3340   (/)c0 3649    Fr wfr 4688   Se wse 4689   Predcpred 27636   TrPredctrpred 27693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-recs 6844  df-rdg 6878  df-pred 27637  df-trpred 27694
This theorem is referenced by:  frind  27716
  Copyright terms: Public domain W3C validator