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Theorem indcardi 8216
Description: Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Hypotheses
Ref Expression
indcardi.a  |-  ( ph  ->  A  e.  V )
indcardi.b  |-  ( ph  ->  T  e.  dom  card )
indcardi.c  |-  ( (
ph  /\  R  ~<_  T  /\  A. y ( S  ~<  R  ->  ch ) )  ->  ps )
indcardi.d  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
indcardi.e  |-  ( x  =  A  ->  ( ps 
<->  th ) )
indcardi.f  |-  ( x  =  y  ->  R  =  S )
indcardi.g  |-  ( x  =  A  ->  R  =  T )
Assertion
Ref Expression
indcardi  |-  ( ph  ->  th )
Distinct variable groups:    x, y, T    x, A    x, S    ch, x    ph, x, y    th, x    y, R    ps, y
Allowed substitution hints:    ps( x)    ch( y)    th( y)    A( y)    R( x)    S( y)    V( x, y)

Proof of Theorem indcardi
StepHypRef Expression
1 indcardi.b . . 3  |-  ( ph  ->  T  e.  dom  card )
2 domrefg 7349 . . 3  |-  ( T  e.  dom  card  ->  T  ~<_  T )
31, 2syl 16 . 2  |-  ( ph  ->  T  ~<_  T )
4 indcardi.a . . 3  |-  ( ph  ->  A  e.  V )
5 cardon 8119 . . . 4  |-  ( card `  T )  e.  On
65a1i 11 . . 3  |-  ( ph  ->  ( card `  T
)  e.  On )
7 simpl1 991 . . . . 5  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) ) )  /\  R  ~<_  T )  ->  ph )
8 simpr 461 . . . . 5  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) ) )  /\  R  ~<_  T )  ->  R  ~<_  T )
9 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<  R )
10 simpl1 991 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ph )
1110, 1syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  T  e.  dom  card )
12 sdomdom 7342 . . . . . . . . . . . . . . . . 17  |-  ( S 
~<  R  ->  S  ~<_  R )
1312adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  R )
14 simpl3 993 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  ~<_  T )
15 domtr 7367 . . . . . . . . . . . . . . . 16  |-  ( ( S  ~<_  R  /\  R  ~<_  T )  ->  S  ~<_  T )
1613, 14, 15syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  T )
17 numdom 8213 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  S  ~<_  T )  ->  S  e.  dom  card )
1811, 16, 17syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  e.  dom  card )
19 numdom 8213 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  R  ~<_  T )  ->  R  e.  dom  card )
2011, 14, 19syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  e.  dom  card )
21 cardsdom2 8163 . . . . . . . . . . . . . 14  |-  ( ( S  e.  dom  card  /\  R  e.  dom  card )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
2218, 20, 21syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
239, 22mpbird 232 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( card `  S
)  e.  ( card `  R ) )
24 id 22 . . . . . . . . . . . . 13  |-  ( ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  (
( card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) ) )
2524com3l 81 . . . . . . . . . . . 12  |-  ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) ) )
2623, 16, 25sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) )
2726ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  ->  ( S  ~<  R  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) ) )
2827com23 78 . . . . . . . . 9  |-  ( (
ph  /\  ( ( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ( S  ~<  R  ->  ch ) ) )
2928alimdv 1675 . . . . . . . 8  |-  ( (
ph  /\  ( ( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  ->  ( A. y
( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  A. y
( S  ~<  R  ->  ch ) ) )
30293exp 1186 . . . . . . 7  |-  ( ph  ->  ( ( ( card `  R )  e.  On  /\  ( card `  R
)  C_  ( card `  T ) )  -> 
( R  ~<_  T  -> 
( A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  A. y
( S  ~<  R  ->  ch ) ) ) ) )
3130com34 83 . . . . . 6  |-  ( ph  ->  ( ( ( card `  R )  e.  On  /\  ( card `  R
)  C_  ( card `  T ) )  -> 
( A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  ( R  ~<_  T  ->  A. y
( S  ~<  R  ->  ch ) ) ) ) )
32313imp1 1200 . . . . 5  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) ) )  /\  R  ~<_  T )  ->  A. y ( S  ~<  R  ->  ch ) )
33 indcardi.c . . . . 5  |-  ( (
ph  /\  R  ~<_  T  /\  A. y ( S  ~<  R  ->  ch ) )  ->  ps )
347, 8, 32, 33syl3anc 1218 . . . 4  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) ) )  /\  R  ~<_  T )  ->  ps )
3534ex 434 . . 3  |-  ( (
ph  /\  ( ( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  A. y ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) ) )  -> 
( R  ~<_  T  ->  ps ) )
36 indcardi.f . . . . 5  |-  ( x  =  y  ->  R  =  S )
3736breq1d 4307 . . . 4  |-  ( x  =  y  ->  ( R  ~<_  T  <->  S  ~<_  T ) )
38 indcardi.d . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
3937, 38imbi12d 320 . . 3  |-  ( x  =  y  ->  (
( R  ~<_  T  ->  ps )  <->  ( S  ~<_  T  ->  ch ) ) )
40 indcardi.g . . . . 5  |-  ( x  =  A  ->  R  =  T )
4140breq1d 4307 . . . 4  |-  ( x  =  A  ->  ( R  ~<_  T  <->  T  ~<_  T ) )
42 indcardi.e . . . 4  |-  ( x  =  A  ->  ( ps 
<->  th ) )
4341, 42imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( R  ~<_  T  ->  ps )  <->  ( T  ~<_  T  ->  th ) ) )
4436fveq2d 5700 . . 3  |-  ( x  =  y  ->  ( card `  R )  =  ( card `  S
) )
4540fveq2d 5700 . . 3  |-  ( x  =  A  ->  ( card `  R )  =  ( card `  T
) )
464, 6, 35, 39, 43, 44, 45tfisi 6474 . 2  |-  ( ph  ->  ( T  ~<_  T  ->  th ) )
473, 46mpd 15 1  |-  ( ph  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756    C_ wss 3333   class class class wbr 4297   Oncon0 4724   dom cdm 4845   ` cfv 5423    ~<_ cdom 7313    ~< csdm 7314   cardccrd 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-recs 6837  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-card 8114
This theorem is referenced by:  uzindi  11808  symggen  15981
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