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Theorem frind 28886
Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 28885). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is itself equal to  A. Compare wfi 28850 and tfi 6659, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
frind  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
Distinct variable groups:    y, A    y, B    y, R

Proof of Theorem frind
StepHypRef Expression
1 ssdif0 3878 . . . . . . 7  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2721 . . . . . 6  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 difss 3624 . . . . . . 7  |-  ( A 
\  B )  C_  A
4 frmin 28885 . . . . . . . . 9  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( ( A  \  B )  C_  A  /\  ( A  \  B
)  =/=  (/) ) )  ->  E. y  e.  ( A  \  B )
Pred ( R , 
( A  \  B
) ,  y )  =  (/) )
5 eldif 3479 . . . . . . . . . . . . 13  |-  ( y  e.  ( A  \  B )  <->  ( y  e.  A  /\  -.  y  e.  B ) )
65anbi1i 695 . . . . . . . . . . . 12  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( ( y  e.  A  /\  -.  y  e.  B )  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) )
7 anass 649 . . . . . . . . . . . 12  |-  ( ( ( y  e.  A  /\  -.  y  e.  B
)  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( y  e.  A  /\  ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) ) )
8 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  ( A 
\  B ) ,  y )  =  (/)  /\ 
-.  y  e.  B
) )
9 indif2 3734 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' R " { y } )  i^i  ( A  \  B ) )  =  ( ( ( `' R " { y } )  i^i  A
)  \  B )
10 df-pred 28807 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( A  \  B )  i^i  ( `' R " { y } ) )
11 incom 3684 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  \  B )  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
1210, 11eqtri 2489 . . . . . . . . . . . . . . . . . 18  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
13 df-pred 28807 . . . . . . . . . . . . . . . . . . . 20  |-  Pred ( R ,  A , 
y )  =  ( A  i^i  ( `' R " { y } ) )
14 incom 3684 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  A
)
1513, 14eqtri 2489 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  A , 
y )  =  ( ( `' R " { y } )  i^i  A )
1615difeq1i 3611 . . . . . . . . . . . . . . . . . 18  |-  ( Pred ( R ,  A ,  y )  \  B )  =  ( ( ( `' R " { y } )  i^i  A )  \  B )
179, 12, 163eqtr4i 2499 . . . . . . . . . . . . . . . . 17  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( Pred ( R ,  A ,  y )  \  B )
1817eqeq1i 2467 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
19 ssdif0 3878 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  A ,  y )  C_  B 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
2018, 19bitr4i 252 . . . . . . . . . . . . . . 15  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<-> 
Pred ( R ,  A ,  y )  C_  B )
2120anbi1i 695 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R , 
( A  \  B
) ,  y )  =  (/)  /\  -.  y  e.  B )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
228, 21bitri 249 . . . . . . . . . . . . 13  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
2322anbi2i 694 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) )  <->  ( y  e.  A  /\  ( Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B ) ) )
246, 7, 233bitri 271 . . . . . . . . . . 11  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( y  e.  A  /\  ( Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B ) ) )
2524rexbii2 2956 . . . . . . . . . 10  |-  ( E. y  e.  ( A 
\  B ) Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  E. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B ) )
26 rexanali 2910 . . . . . . . . . 10  |-  ( E. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B )  <->  -.  A. y  e.  A  ( Pred ( R ,  A , 
y )  C_  B  ->  y  e.  B ) )
2725, 26bitri 249 . . . . . . . . 9  |-  ( E. y  e.  ( A 
\  B ) Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  -.  A. y  e.  A  ( Pred ( R ,  A , 
y )  C_  B  ->  y  e.  B ) )
284, 27sylib 196 . . . . . . . 8  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( ( A  \  B )  C_  A  /\  ( A  \  B
)  =/=  (/) ) )  ->  -.  A. y  e.  A  ( Pred ( R ,  A , 
y )  C_  B  ->  y  e.  B ) )
2928ex 434 . . . . . . 7  |-  ( ( R  Fr  A  /\  R Se  A )  ->  (
( ( A  \  B )  C_  A  /\  ( A  \  B
)  =/=  (/) )  ->  -.  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
303, 29mpani 676 . . . . . 6  |-  ( ( R  Fr  A  /\  R Se  A )  ->  (
( A  \  B
)  =/=  (/)  ->  -.  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
312, 30syl5bi 217 . . . . 5  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( -.  A  C_  B  ->  -.  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
3231con4d 105 . . . 4  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B )  ->  A  C_  B ) )
3332imp 429 . . 3  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  C_  B )
3433adantrl 715 . 2  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  C_  B
)
35 simprl 755 . 2  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  B  C_  A
)
3634, 35eqssd 3514 1  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808    \ cdif 3466    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020    Fr wfr 4828   Se wse 4829   `'ccnv 4991   "cima 4995   Predcpred 28806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066  df-pred 28807  df-trpred 28864
This theorem is referenced by:  frindi  28887  frinsg  28888
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