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Theorem tfi 4602
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 4611 or tfinds 4608 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 3260 . . . . . . . . 9  |-  ( x  e.  ( On  \  A )  ->  -.  x  e.  A )
21adantl 454 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  x  e.  A )
3 eldifi 3259 . . . . . . . . . 10  |-  ( x  e.  ( On  \  A )  ->  x  e.  On )
4 onss 4540 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  x  C_  On )
5 difin0ss 3481 . . . . . . . . . . . . 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 14 . . . . . . . . . 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 420 . . . . . . . 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 425 . . . . . 6  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  -.  (
( On  \  A
)  i^i  x )  =  (/) ) )
1312ralimi2 2588 . . . . 5  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  A. x  e.  ( On  \  A )  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
14 ralnex 2526 . . . . 5  |-  ( A. x  e.  ( On  \  A )  -.  (
( On  \  A
)  i^i  x )  =  (/)  <->  -.  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
1513, 14sylib 190 . . . 4  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  -.  E. x  e.  ( On  \  A ) ( ( On  \  A )  i^i  x
)  =  (/) )
16 ssdif0 3474 . . . . . 6  |-  ( On  C_  A  <->  ( On  \  A )  =  (/) )
1716necon3bbii 2450 . . . . 5  |-  ( -.  On  C_  A  <->  ( On  \  A )  =/=  (/) )
18 ordon 4532 . . . . . 6  |-  Ord  On
19 difss 3264 . . . . . 6  |-  ( On 
\  A )  C_  On
20 tz7.5 4371 . . . . . 6  |-  ( ( Ord  On  /\  ( On  \  A )  C_  On  /\  ( On  \  A )  =/=  (/) )  ->  E. x  e.  ( On  \  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2118, 19, 20mp3an12 1272 . . . . 5  |-  ( ( On  \  A )  =/=  (/)  ->  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
2217, 21sylbi 189 . . . 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 555 . 2  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  ( A  C_  On  /\  On  C_  A
) )
25 eqss 3155 . 2  |-  ( A  =  On  <->  ( A  C_  On  /\  On  C_  A ) )
2624, 25sylibr 205 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 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    \ cdif 3110    i^i cin 3112    C_ wss 3113   (/)c0 3416   Ord word 4349   Oncon0 4350
This theorem is referenced by:  tfis  4603  tfisg  23559
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-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354
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