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Theorem tfi 4833
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 4842 or tfinds 4839 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 3470 . . . . . . . . 9  |-  ( x  e.  ( On  \  A )  ->  -.  x  e.  A )
21adantl 453 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  x  e.  A )
3 eldifi 3469 . . . . . . . . . 10  |-  ( x  e.  ( On  \  A )  ->  x  e.  On )
4 onss 4771 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  x  C_  On )
5 difin0ss 3694 . . . . . . . . . . . . 13  |-  ( ( ( On  \  A
)  i^i  x )  =  (/)  ->  ( x  C_  On  ->  x  C_  A
) )
64, 5syl5com 28 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  C_  A ) )
76imim1d 71 . . . . . . . . . . 11  |-  ( x  e.  On  ->  (
( x  C_  A  ->  x  e.  A )  ->  ( ( ( On  \  A )  i^i  x )  =  (/)  ->  x  e.  A
) ) )
87a2i 13 . . . . . . . . . 10  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  On  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
93, 8syl5 30 . . . . . . . . 9  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
109imp 419 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  e.  A ) )
112, 10mtod 170 . . . . . . 7  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
1211ex 424 . . . . . 6  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  -.  (
( On  \  A
)  i^i  x )  =  (/) ) )
1312ralimi2 2778 . . . . 5  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  A. x  e.  ( On  \  A )  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
14 ralnex 2715 . . . . 5  |-  ( A. x  e.  ( On  \  A )  -.  (
( On  \  A
)  i^i  x )  =  (/)  <->  -.  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
1513, 14sylib 189 . . . 4  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  -.  E. x  e.  ( On  \  A ) ( ( On  \  A )  i^i  x
)  =  (/) )
16 ssdif0 3686 . . . . . 6  |-  ( On  C_  A  <->  ( On  \  A )  =  (/) )
1716necon3bbii 2632 . . . . 5  |-  ( -.  On  C_  A  <->  ( On  \  A )  =/=  (/) )
18 ordon 4763 . . . . . 6  |-  Ord  On
19 difss 3474 . . . . . 6  |-  ( On 
\  A )  C_  On
20 tz7.5 4602 . . . . . 6  |-  ( ( Ord  On  /\  ( On  \  A )  C_  On  /\  ( On  \  A )  =/=  (/) )  ->  E. x  e.  ( On  \  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2118, 19, 20mp3an12 1269 . . . . 5  |-  ( ( On  \  A )  =/=  (/)  ->  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
2217, 21sylbi 188 . . . 4  |-  ( -.  On  C_  A  ->  E. x  e.  ( On 
\  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2315, 22nsyl2 121 . . 3  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  On  C_  A )
2423anim2i 553 . 2  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  ( A  C_  On  /\  On  C_  A
) )
25 eqss 3363 . 2  |-  ( A  =  On  <->  ( A  C_  On  /\  On  C_  A ) )
2624, 25sylibr 204 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   Ord word 4580   Oncon0 4581
This theorem is referenced by:  tfis  4834  tfisg  25479
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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