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Theorem frind 30054
Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 30053). 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 5400 and tfi 6671, 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 3828 . . . . . . 7  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2664 . . . . . 6  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 difss 3570 . . . . . . 7  |-  ( A 
\  B )  C_  A
4 frmin 30053 . . . . . . . . 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 3424 . . . . . . . . . . . . 13  |-  ( y  e.  ( A  \  B )  <->  ( y  e.  A  /\  -.  y  e.  B ) )
65anbi1i 693 . . . . . . . . . . . 12  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( ( y  e.  A  /\  -.  y  e.  B )  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) )
7 anass 647 . . . . . . . . . . . 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 448 . . . . . . . . . . . . . 14  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  ( A 
\  B ) ,  y )  =  (/)  /\ 
-.  y  e.  B
) )
9 indif2 3693 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' R " { y } )  i^i  ( A  \  B ) )  =  ( ( ( `' R " { y } )  i^i  A
)  \  B )
10 df-pred 5367 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( A  \  B )  i^i  ( `' R " { y } ) )
11 incom 3632 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  \  B )  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
1210, 11eqtri 2431 . . . . . . . . . . . . . . . . . 18  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
13 df-pred 5367 . . . . . . . . . . . . . . . . . . . 20  |-  Pred ( R ,  A , 
y )  =  ( A  i^i  ( `' R " { y } ) )
14 incom 3632 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  A
)
1513, 14eqtri 2431 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  A , 
y )  =  ( ( `' R " { y } )  i^i  A )
1615difeq1i 3557 . . . . . . . . . . . . . . . . . 18  |-  ( Pred ( R ,  A ,  y )  \  B )  =  ( ( ( `' R " { y } )  i^i  A )  \  B )
179, 12, 163eqtr4i 2441 . . . . . . . . . . . . . . . . 17  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( Pred ( R ,  A ,  y )  \  B )
1817eqeq1i 2409 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
19 ssdif0 3828 . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . 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 2904 . . . . . . . . . 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 2857 . . . . . . . . . 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 432 . . . . . . 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 674 . . . . . 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 427 . . 3  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  C_  B )
3433adantrl 714 . 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 756 . 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 3459 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738   {csn 3972    Fr wfr 4779   Se wse 4780   `'ccnv 4822   "cima 4826   Predcpred 5366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-trpred 30032
This theorem is referenced by:  frindi  30055  frinsg  30056
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