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Theorem tfi 6575
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 6584 or tfinds 6581 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 3588 . . . . . . . . 9  |-  ( x  e.  ( On  \  A )  ->  -.  x  e.  A )
21adantl 466 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  x  e.  A )
3 eldifi 3587 . . . . . . . . . 10  |-  ( x  e.  ( On  \  A )  ->  x  e.  On )
4 onss 6513 . . . . . . . . . . . . 13  |-  ( x  e.  On  ->  x  C_  On )
5 difin0ss 3854 . . . . . . . . . . . . 13  |-  ( ( ( On  \  A
)  i^i  x )  =  (/)  ->  ( x  C_  On  ->  x  C_  A
) )
64, 5syl5com 30 . . . . . . . . . . . 12  |-  ( x  e.  On  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  C_  A ) )
76imim1d 75 . . . . . . . . . . 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 32 . . . . . . . . 9  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  ( (
( On  \  A
)  i^i  x )  =  (/)  ->  x  e.  A ) ) )
109imp 429 . . . . . . . 8  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  (
( ( On  \  A )  i^i  x
)  =  (/)  ->  x  e.  A ) )
112, 10mtod 177 . . . . . . 7  |-  ( ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  /\  x  e.  ( On  \  A
) )  ->  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
1211ex 434 . . . . . 6  |-  ( ( x  e.  On  ->  ( x  C_  A  ->  x  e.  A ) )  ->  ( x  e.  ( On  \  A
)  ->  -.  (
( On  \  A
)  i^i  x )  =  (/) ) )
1312ralimi2 2816 . . . . 5  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  A. x  e.  ( On  \  A )  -.  ( ( On  \  A )  i^i  x
)  =  (/) )
14 ralnex 2852 . . . . 5  |-  ( A. x  e.  ( On  \  A )  -.  (
( On  \  A
)  i^i  x )  =  (/)  <->  -.  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
1513, 14sylib 196 . . . 4  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  -.  E. x  e.  ( On  \  A ) ( ( On  \  A )  i^i  x
)  =  (/) )
16 ssdif0 3846 . . . . . 6  |-  ( On  C_  A  <->  ( On  \  A )  =  (/) )
1716necon3bbii 2713 . . . . 5  |-  ( -.  On  C_  A  <->  ( On  \  A )  =/=  (/) )
18 ordon 6505 . . . . . 6  |-  Ord  On
19 difss 3592 . . . . . 6  |-  ( On 
\  A )  C_  On
20 tz7.5 4849 . . . . . 6  |-  ( ( Ord  On  /\  ( On  \  A )  C_  On  /\  ( On  \  A )  =/=  (/) )  ->  E. x  e.  ( On  \  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2118, 19, 20mp3an12 1305 . . . . 5  |-  ( ( On  \  A )  =/=  (/)  ->  E. x  e.  ( On  \  A
) ( ( On 
\  A )  i^i  x )  =  (/) )
2217, 21sylbi 195 . . . 4  |-  ( -.  On  C_  A  ->  E. x  e.  ( On 
\  A ) ( ( On  \  A
)  i^i  x )  =  (/) )
2315, 22nsyl2 127 . . 3  |-  ( A. x  e.  On  (
x  C_  A  ->  x  e.  A )  ->  On  C_  A )
2423anim2i 569 . 2  |-  ( ( A  C_  On  /\  A. x  e.  On  (
x  C_  A  ->  x  e.  A ) )  ->  ( A  C_  On  /\  On  C_  A
) )
25 eqss 3480 . 2  |-  ( A  =  On  <->  ( A  C_  On  /\  On  C_  A ) )
2624, 25sylibr 212 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 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3746   Ord word 4827   Oncon0 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832
This theorem is referenced by:  tfis  6576  tfisg  27810
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