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Theorem indcardi 7848
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 7071 . . 3  |-  ( T  e.  dom  card  ->  T  ~<_  T )
31, 2syl 16 . 2  |-  ( ph  ->  T  ~<_  T )
4 indcardi.a . . 3  |-  ( ph  ->  A  e.  V )
5 cardon 7757 . . . 4  |-  ( card `  T )  e.  On
65a1i 11 . . 3  |-  ( ph  ->  ( card `  T
)  e.  On )
7 simpl1 960 . . . . 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 448 . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<  R )
10 simpl1 960 . . . . . . . . . . . . . . . 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 7064 . . . . . . . . . . . . . . . . 17  |-  ( S 
~<  R  ->  S  ~<_  R )
1312adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  R )
14 simpl3 962 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  ~<_  T )
15 domtr 7089 . . . . . . . . . . . . . . . 16  |-  ( ( S  ~<_  R  /\  R  ~<_  T )  ->  S  ~<_  T )
1613, 14, 15syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  T )
17 numdom 7845 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  S  ~<_  T )  ->  S  e.  dom  card )
1811, 16, 17syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  e.  dom  card )
19 numdom 7845 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  R  ~<_  T )  ->  R  e.  dom  card )
2011, 14, 19syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  e.  dom  card )
21 cardsdom2 7801 . . . . . . . . . . . . . 14  |-  ( ( S  e.  dom  card  /\  R  e.  dom  card )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
2218, 20, 21syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
239, 22mpbird 224 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( card `  S
)  e.  ( card `  R ) )
24 id 20 . . . . . . . . . . . . 13  |-  ( ( ( card `  S
)  e.  ( card `  R )  ->  ( S  ~<_  T  ->  ch ) )  ->  (
( card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) ) )
2524com3l 77 . . . . . . . . . . . 12  |-  ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) ) )
2623, 16, 25sylc 58 . . . . . . . . . . 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 424 . . . . . . . . . 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 74 . . . . . . . . 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 1628 . . . . . . . 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 1152 . . . . . . 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 79 . . . . . 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 1166 . . . . 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 1184 . . . 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 424 . . 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 4156 . . . 4  |-  ( x  =  y  ->  ( R  ~<_  T  <->  S  ~<_  T ) )
38 indcardi.d . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
3937, 38imbi12d 312 . . 3  |-  ( x  =  y  ->  (
( R  ~<_  T  ->  ps )  <->  ( S  ~<_  T  ->  ch ) ) )
40 indcardi.g . . . . 5  |-  ( x  =  A  ->  R  =  T )
4140breq1d 4156 . . . 4  |-  ( x  =  A  ->  ( R  ~<_  T  <->  T  ~<_  T ) )
42 indcardi.e . . . 4  |-  ( x  =  A  ->  ( ps 
<->  th ) )
4341, 42imbi12d 312 . . 3  |-  ( x  =  A  ->  (
( R  ~<_  T  ->  ps )  <->  ( T  ~<_  T  ->  th ) ) )
4436fveq2d 5665 . . 3  |-  ( x  =  y  ->  ( card `  R )  =  ( card `  S
) )
4540fveq2d 5665 . . 3  |-  ( x  =  A  ->  ( card `  R )  =  ( card `  T
) )
464, 6, 35, 39, 43, 44, 45tfisi 4771 . 2  |-  ( ph  ->  ( T  ~<_  T  ->  th ) )
473, 46mpd 15 1  |-  ( ph  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1717    C_ wss 3256   class class class wbr 4146   Oncon0 4515   dom cdm 4811   ` cfv 5387    ~<_ cdom 7036    ~< csdm 7037   cardccrd 7748
This theorem is referenced by:  uzindi  11240  symggen  27073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-riota 6478  df-recs 6562  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-card 7752
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