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Theorem wfi 25462
Description: The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if  B is a subclass of a well-ordered class  A with the property that every element of  B whose inital segment is included in 
A is itself equal to  A. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
wfi  |-  ( ( ( R  We  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 wfi
StepHypRef Expression
1 ssdif0 3678 . . . . . . 7  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2629 . . . . . 6  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 difss 3466 . . . . . . 7  |-  ( A 
\  B )  C_  A
4 tz6.26 25460 . . . . . . . . 9  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( ( A  \  B )  C_  A  /\  ( A  \  B
)  =/=  (/) ) )  ->  E. y  e.  ( A  \  B )
Pred ( R , 
( A  \  B
) ,  y )  =  (/) )
5 eldif 3322 . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' R " { y } )  i^i  ( A  \  B ) )  =  ( ( ( `' R " { y } )  i^i  A
)  \  B )
10 df-pred 25423 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( A  \  B )  i^i  ( `' R " { y } ) )
11 incom 3525 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  \  B )  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
1210, 11eqtri 2455 . . . . . . . . . . . . . . . . . 18  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
13 df-pred 25423 . . . . . . . . . . . . . . . . . . . 20  |-  Pred ( R ,  A , 
y )  =  ( A  i^i  ( `' R " { y } ) )
14 incom 3525 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  A
)
1513, 14eqtri 2455 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  A , 
y )  =  ( ( `' R " { y } )  i^i  A )
1615difeq1i 3453 . . . . . . . . . . . . . . . . . 18  |-  ( Pred ( R ,  A ,  y )  \  B )  =  ( ( ( `' R " { y } )  i^i  A )  \  B )
179, 12, 163eqtr4i 2465 . . . . . . . . . . . . . . . . 17  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( Pred ( R ,  A ,  y )  \  B )
1817eqeq1i 2442 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
19 ssdif0 3678 . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . 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 2743 . . . . . . . . . 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  We  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  We  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  We  A  /\  R Se  A )  ->  (
( A  \  B
)  =/=  (/)  ->  -.  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
312, 30syl5bi 209 . . . . 5  |-  ( ( R  We  A  /\  R Se  A )  ->  ( -.  A  C_  B  ->  -.  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
3231con4d 99 . . . 4  |-  ( ( R  We  A  /\  R Se  A )  ->  ( A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B )  ->  A  C_  B ) )
3332imp 419 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  C_  B )
3433adantrl 697 . 2  |-  ( ( ( R  We  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  We  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 3357 1  |-  ( ( ( R  We  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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   Se wse 4531    We wwe 4532   `'ccnv 4868   "cima 4872   Predcpred 25422
This theorem is referenced by:  wfii  25463  wfisg  25464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-pred 25423
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