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Theorem tfi 6638
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 6647 or tfinds 6644 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 3531 . . . . . . . . 9  |-  ( x  e.  ( On  \  A )  ->  -.  x  e.  A )
21adantl 467 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  x  e.  A )
3 eldifi 3530 . . . . . . . . . 10  |-  ( x  e.  ( On  \  A )  ->  x  e.  On )
4 onss 6575 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  x  C_  On )
5 difin0ss 3806 . . . . . . . . . . . . 13  |-  ( ( ( On  \  A
)  i^i  x )  =  (/)  ->  ( x  C_  On  ->  x  C_  A
) )
64, 5syl5com 31 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  C_  A ) )
76imim1d 78 . . . . . . . . . . 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 33 . . . . . . . . 9  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
109imp 430 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  e.  A ) )
112, 10mtod 180 . . . . . . 7  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
1211ex 435 . . . . . 6  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  -.  (
( On  \  A
)  i^i  x )  =  (/) ) )
1312ralimi2 2755 . . . . 5  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  A. x  e.  ( On  \  A )  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
14 ralnex 2811 . . . . 5  |-  ( A. x  e.  ( On  \  A )  -.  (
( On  \  A
)  i^i  x )  =  (/)  <->  -.  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
1513, 14sylib 199 . . . 4  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  -.  E. x  e.  ( On  \  A ) ( ( On  \  A )  i^i  x
)  =  (/) )
16 ssdif0 3796 . . . . . 6  |-  ( On  C_  A  <->  ( On  \  A )  =  (/) )
1716necon3bbii 2648 . . . . 5  |-  ( -.  On  C_  A  <->  ( On  \  A )  =/=  (/) )
18 ordon 6567 . . . . . 6  |-  Ord  On
19 difss 3535 . . . . . 6  |-  ( On 
\  A )  C_  On
20 tz7.5 5406 . . . . . 6  |-  ( ( Ord  On  /\  ( On  \  A )  C_  On  /\  ( On  \  A )  =/=  (/) )  ->  E. x  e.  ( On  \  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2118, 19, 20mp3an12 1350 . . . . 5  |-  ( ( On  \  A )  =/=  (/)  ->  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
2217, 21sylbi 198 . . . 4  |-  ( -.  On  C_  A  ->  E. x  e.  ( On 
\  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2315, 22nsyl2 130 . . 3  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  On  C_  A )
2423anim2i 571 . 2  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  ( A  C_  On  /\  On  C_  A
) )
25 eqss 3422 . 2  |-  ( A  =  On  <->  ( A  C_  On  /\  On  C_  A ) )
2624, 25sylibr 215 1  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715    \ cdif 3376    i^i cin 3378    C_ wss 3379   (/)c0 3704   Ord word 5384   Oncon0 5385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389
This theorem is referenced by:  tfis  6639  tfisg  30408
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