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Theorem indcardi 8439
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 7569 . . 3  |-  ( T  e.  dom  card  ->  T  ~<_  T )
31, 2syl 16 . 2  |-  ( ph  ->  T  ~<_  T )
4 indcardi.a . . 3  |-  ( ph  ->  A  e.  V )
5 cardon 8342 . . . 4  |-  ( card `  T )  e.  On
65a1i 11 . . 3  |-  ( ph  ->  ( card `  T
)  e.  On )
7 simpl1 999 . . . . 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 999 . . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  ~<_  T )
15 domtr 7587 . . . . . . . . . . . . . . . 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 8436 . . . . . . . . . . . . . . 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 8436 . . . . . . . . . . . . . . 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 8386 . . . . . . . . . . . . . 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 1710 . . . . . . . 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 1195 . . . . . . 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 1209 . . . . 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 1228 . . . 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 4466 . . . 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 4466 . . . 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 5876 . . 3  |-  ( x  =  y  ->  ( card `  R )  =  ( card `  S
) )
4540fveq2d 5876 . . 3  |-  ( x  =  A  ->  ( card `  R )  =  ( card `  T
) )
464, 6, 35, 39, 43, 44, 45tfisi 6692 . 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 973   A.wal 1393    = wceq 1395    e. wcel 1819    C_ wss 3471   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ` cfv 5594    ~<_ cdom 7533    ~< csdm 7534   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-card 8337
This theorem is referenced by:  uzindi  12094  symggen  16622
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