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Theorem tfi 4660
Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if  A is a class of ordinal numbers with the property that every ordinal number included in  A also belongs to  A, then every ordinal number is in  A.

See theorem tfindes 4669 or tfinds 4666 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Assertion
Ref Expression
tfi  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
Distinct variable group:    x, A

Proof of Theorem tfi
StepHypRef Expression
1 eldifn 3312 . . . . . . . . 9  |-  ( x  e.  ( On  \  A )  ->  -.  x  e.  A )
21adantl 452 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  x  e.  A )
3 eldifi 3311 . . . . . . . . . 10  |-  ( x  e.  ( On  \  A )  ->  x  e.  On )
4 onss 4598 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  x  C_  On )
5 difin0ss 3533 . . . . . . . . . . . . 13  |-  ( ( ( On  \  A
)  i^i  x )  =  (/)  ->  ( x  C_  On  ->  x  C_  A
) )
64, 5syl5com 26 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  C_  A ) )
76imim1d 69 . . . . . . . . . . 11  |-  ( x  e.  On  ->  (
( x  C_  A  ->  x  e.  A )  ->  ( ( ( On  \  A )  i^i  x )  =  (/)  ->  x  e.  A
) ) )
87a2i 12 . . . . . . . . . 10  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  On  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
93, 8syl5 28 . . . . . . . . 9  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
109imp 418 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  e.  A ) )
112, 10mtod 168 . . . . . . 7  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
1211ex 423 . . . . . 6  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  -.  (
( On  \  A
)  i^i  x )  =  (/) ) )
1312ralimi2 2628 . . . . 5  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  A. x  e.  ( On  \  A )  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
14 ralnex 2566 . . . . 5  |-  ( A. x  e.  ( On  \  A )  -.  (
( On  \  A
)  i^i  x )  =  (/)  <->  -.  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
1513, 14sylib 188 . . . 4  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  -.  E. x  e.  ( On  \  A ) ( ( On  \  A )  i^i  x
)  =  (/) )
16 ssdif0 3526 . . . . . 6  |-  ( On  C_  A  <->  ( On  \  A )  =  (/) )
1716necon3bbii 2490 . . . . 5  |-  ( -.  On  C_  A  <->  ( On  \  A )  =/=  (/) )
18 ordon 4590 . . . . . 6  |-  Ord  On
19 difss 3316 . . . . . 6  |-  ( On 
\  A )  C_  On
20 tz7.5 4429 . . . . . 6  |-  ( ( Ord  On  /\  ( On  \  A )  C_  On  /\  ( On  \  A )  =/=  (/) )  ->  E. x  e.  ( On  \  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2118, 19, 20mp3an12 1267 . . . . 5  |-  ( ( On  \  A )  =/=  (/)  ->  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
2217, 21sylbi 187 . . . 4  |-  ( -.  On  C_  A  ->  E. x  e.  ( On 
\  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2315, 22nsyl2 119 . . 3  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  On  C_  A )
2423anim2i 552 . 2  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  ( A  C_  On  /\  On  C_  A
) )
25 eqss 3207 . 2  |-  ( A  =  On  <->  ( A  C_  On  /\  On  C_  A ) )
2624, 25sylibr 203 1  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   Ord word 4407   Oncon0 4408
This theorem is referenced by:  tfis  4661  tfisg  24275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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