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Theorem indcardi 7668
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 6896 . . 3  |-  ( T  e.  dom  card  ->  T  ~<_  T )
31, 2syl 15 . 2  |-  ( ph  ->  T  ~<_  T )
4 indcardi.a . . 3  |-  ( ph  ->  A  e.  V )
5 cardon 7577 . . . 4  |-  ( card `  T )  e.  On
65a1i 10 . . 3  |-  ( ph  ->  ( card `  T
)  e.  On )
7 simpl1 958 . . . . 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 447 . . . . 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 447 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<  R )
10 simpl1 958 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ph )
1110, 1syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  T  e.  dom  card )
12 sdomdom 6889 . . . . . . . . . . . . . . . . 17  |-  ( S 
~<  R  ->  S  ~<_  R )
1312adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  R )
14 simpl3 960 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  ~<_  T )
15 domtr 6914 . . . . . . . . . . . . . . . 16  |-  ( ( S  ~<_  R  /\  R  ~<_  T )  ->  S  ~<_  T )
1613, 14, 15syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  T )
17 numdom 7665 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  S  ~<_  T )  ->  S  e.  dom  card )
1811, 16, 17syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  e.  dom  card )
19 numdom 7665 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  R  ~<_  T )  ->  R  e.  dom  card )
2011, 14, 19syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  e.  dom  card )
21 cardsdom2 7621 . . . . . . . . . . . . . 14  |-  ( ( S  e.  dom  card  /\  R  e.  dom  card )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
2218, 20, 21syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
239, 22mpbird 223 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( card `  S
)  e.  ( card `  R ) )
24 id 19 . . . . . . . . . . . . 13  |-  ( ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  (
( card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) ) )
2524com3l 75 . . . . . . . . . . . 12  |-  ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) ) )
2623, 16, 25sylc 56 . . . . . . . . . . 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 423 . . . . . . . . . 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 72 . . . . . . . . 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 1607 . . . . . . . 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 1150 . . . . . . 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 77 . . . . . 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 1164 . . . . 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 1182 . . . 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 423 . . 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 4033 . . . 4  |-  ( x  =  y  ->  ( R  ~<_  T  <->  S  ~<_  T ) )
38 indcardi.d . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
3937, 38imbi12d 311 . . 3  |-  ( x  =  y  ->  (
( R  ~<_  T  ->  ps )  <->  ( S  ~<_  T  ->  ch ) ) )
40 indcardi.g . . . . 5  |-  ( x  =  A  ->  R  =  T )
4140breq1d 4033 . . . 4  |-  ( x  =  A  ->  ( R  ~<_  T  <->  T  ~<_  T ) )
42 indcardi.e . . . 4  |-  ( x  =  A  ->  ( ps 
<->  th ) )
4341, 42imbi12d 311 . . 3  |-  ( x  =  A  ->  (
( R  ~<_  T  ->  ps )  <->  ( T  ~<_  T  ->  th ) ) )
4436fveq2d 5529 . . 3  |-  ( x  =  y  ->  ( card `  R )  =  ( card `  S
) )
4540fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( card `  R )  =  ( card `  T
) )
464, 6, 35, 39, 43, 44, 45tfisi 4649 . 2  |-  ( ph  ->  ( T  ~<_  T  ->  th ) )
473, 46mpd 14 1  |-  ( ph  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  uzindi  11043  symggen  26823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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