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

Theorem wfi 23375
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 3420 . . . . . . 7  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
21necon3bbii 2443 . . . . . 6  |-  ( -.  A  C_  B  <->  ( A  \  B )  =/=  (/) )
3 difss 3220 . . . . . . 7  |-  ( A 
\  B )  C_  A
4 tz6.26 23373 . . . . . . . . 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 3088 . . . . . . . . . . . . 13  |-  ( y  e.  ( A  \  B )  <->  ( y  e.  A  /\  -.  y  e.  B ) )
65anbi1i 679 . . . . . . . . . . . 12  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( ( y  e.  A  /\  -.  y  e.  B )  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) ) )
7 anass 633 . . . . . . . . . . . 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 439 . . . . . . . . . . . . . 14  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  ( A 
\  B ) ,  y )  =  (/)  /\ 
-.  y  e.  B
) )
9 indif2 3319 . . . . . . . . . . . . . . . . . 18  |-  ( ( `' R " { y } )  i^i  ( A  \  B ) )  =  ( ( ( `' R " { y } )  i^i  A
)  \  B )
10 df-pred 23336 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( A  \  B )  i^i  ( `' R " { y } ) )
11 incom 3269 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  \  B )  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
1210, 11eqtri 2273 . . . . . . . . . . . . . . . . . 18  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( ( `' R " { y } )  i^i  ( A  \  B ) )
13 df-pred 23336 . . . . . . . . . . . . . . . . . . . 20  |-  Pred ( R ,  A , 
y )  =  ( A  i^i  ( `' R " { y } ) )
14 incom 3269 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  i^i  ( `' R " { y } ) )  =  ( ( `' R " { y } )  i^i  A
)
1513, 14eqtri 2273 . . . . . . . . . . . . . . . . . . 19  |-  Pred ( R ,  A , 
y )  =  ( ( `' R " { y } )  i^i  A )
1615difeq1i 3207 . . . . . . . . . . . . . . . . . 18  |-  ( Pred ( R ,  A ,  y )  \  B )  =  ( ( ( `' R " { y } )  i^i  A )  \  B )
179, 12, 163eqtr4i 2283 . . . . . . . . . . . . . . . . 17  |-  Pred ( R ,  ( A  \  B ) ,  y )  =  ( Pred ( R ,  A ,  y )  \  B )
1817eqeq1i 2260 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
19 ssdif0 3420 . . . . . . . . . . . . . . . 16  |-  ( Pred ( R ,  A ,  y )  C_  B 
<->  ( Pred ( R ,  A ,  y )  \  B )  =  (/) )
2018, 19bitr4i 245 . . . . . . . . . . . . . . 15  |-  ( Pred ( R ,  ( A  \  B ) ,  y )  =  (/) 
<-> 
Pred ( R ,  A ,  y )  C_  B )
2120anbi1i 679 . . . . . . . . . . . . . 14  |-  ( (
Pred ( R , 
( A  \  B
) ,  y )  =  (/)  /\  -.  y  e.  B )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
228, 21bitri 242 . . . . . . . . . . . . 13  |-  ( ( -.  y  e.  B  /\  Pred ( R , 
( A  \  B
) ,  y )  =  (/) )  <->  ( Pred ( R ,  A , 
y )  C_  B  /\  -.  y  e.  B
) )
2322anbi2i 678 . . . . . . . . . . . 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 264 . . . . . . . . . . 11  |-  ( ( y  e.  ( A 
\  B )  /\  Pred ( R ,  ( A  \  B ) ,  y )  =  (/) )  <->  ( y  e.  A  /\  ( Pred ( R ,  A ,  y )  C_  B  /\  -.  y  e.  B ) ) )
2524rexbii2 2536 . . . . . . . . . 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 2551 . . . . . . . . . 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 242 . . . . . . . . 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 190 . . . . . . . 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 425 . . . . . . 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 660 . . . . . 6  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( A  \  B
)  =/=  (/)  ->  -.  A. y  e.  A  (
Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )
312, 30syl5bi 210 . . . . 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 420 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  C_  B )
3433adantrl 699 . 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 735 . 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 3117 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 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510    \ cdif 3075    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   Se wse 4243    We wwe 4244   `'ccnv 4579   "cima 4583   Predcpred 23335
This theorem is referenced by:  wfii  23376  wfisg  23377
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-pred 23336
  Copyright terms: Public domain W3C validator