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Theorem frind 25465
Description: The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 25464). 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 25429 and tfi 4800, 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 3654 . . . . . . 7  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2606 . . . . . 6  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 difss 3442 . . . . . . 7  |-  ( A 
\  B )  C_  A
4 frmin 25464 . . . . . . . . 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 3298 . . . . . . . . . . . . 13  |-  ( y  e.  ( A  \  B )  <->  ( y  e.  A  /\  -.  y  e.  B ) )
65anbi1i 677 . . . . . . . . . . . 12  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( ( y  e.  A  /\  -.  y  e.  B )  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) )
7 anass 631 . . . . . . . . . . . 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 438 . . . . . . . . . . . . . 14  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  ( A 
\  B ) ,  y )  =  (/)  /\ 
-.  y  e.  B
) )
9 indif2 3552 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' R " { y } )  i^i  ( A  \  B ) )  =  ( ( ( `' R " { y } )  i^i  A
)  \  B )
10 df-pred 25390 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( A  \  B )  i^i  ( `' R " { y } ) )
11 incom 3501 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  \  B )  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
1210, 11eqtri 2432 . . . . . . . . . . . . . . . . . 18  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
13 df-pred 25390 . . . . . . . . . . . . . . . . . . . 20  |-  Pred ( R ,  A , 
y )  =  ( A  i^i  ( `' R " { y } ) )
14 incom 3501 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  A
)
1513, 14eqtri 2432 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  A , 
y )  =  ( ( `' R " { y } )  i^i  A )
1615difeq1i 3429 . . . . . . . . . . . . . . . . . 18  |-  ( Pred ( R ,  A ,  y )  \  B )  =  ( ( ( `' R " { y } )  i^i  A )  \  B )
179, 12, 163eqtr4i 2442 . . . . . . . . . . . . . . . . 17  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( Pred ( R ,  A ,  y )  \  B )
1817eqeq1i 2419 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
19 ssdif0 3654 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  A ,  y )  C_  B 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
2018, 19bitr4i 244 . . . . . . . . . . . . . . 15  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<-> 
Pred ( R ,  A ,  y )  C_  B )
2120anbi1i 677 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R , 
( A  \  B
) ,  y )  =  (/)  /\  -.  y  e.  B )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
228, 21bitri 241 . . . . . . . . . . . . 13  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
2322anbi2i 676 . . . . . . . . . . . 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 263 . . . . . . . . . . 11  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( y  e.  A  /\  ( Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B ) ) )
2524rexbii2 2703 . . . . . . . . . 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 2720 . . . . . . . . . 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 241 . . . . . . . . 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 189 . . . . . . . 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 424 . . . . . . 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 658 . . . . . 6  |-  ( ( R  Fr  A  /\  R Se  A )  ->  (
( A  \  B
)  =/=  (/)  ->  -.  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
312, 30syl5bi 209 . . . . 5  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( -.  A  C_  B  ->  -.  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
3231con4d 99 . . . 4  |-  ( ( R  Fr  A  /\  R Se  A )  ->  ( A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B )  ->  A  C_  B ) )
3332imp 419 . . 3  |-  ( ( ( R  Fr  A  /\  R Se  A )  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  C_  B )
3433adantrl 697 . 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 733 . 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 3333 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675    \ cdif 3285    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782    Fr wfr 4506   Se wse 4507   `'ccnv 4844   "cima 4848   Predcpred 25389
This theorem is referenced by:  frindi  25466  frinsg  25467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-pred 25390  df-trpred 25443
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