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Theorem indcardi 8477
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 7609 . . 3  |-  ( T  e.  dom  card  ->  T  ~<_  T )
31, 2syl 17 . 2  |-  ( ph  ->  T  ~<_  T )
4 indcardi.a . . 3  |-  ( ph  ->  A  e.  V )
5 cardon 8383 . . . 4  |-  ( card `  T )  e.  On
65a1i 11 . . 3  |-  ( ph  ->  ( card `  T
)  e.  On )
7 simpl1 1012 . . . . 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 463 . . . . 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 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<  R )
10 simpl1 1012 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ph )
1110, 1syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  T  e.  dom  card )
12 sdomdom 7602 . . . . . . . . . . . . . . . . 17  |-  ( S 
~<  R  ->  S  ~<_  R )
1312adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  R )
14 simpl3 1014 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  ~<_  T )
15 domtr 7627 . . . . . . . . . . . . . . . 16  |-  ( ( S  ~<_  R  /\  R  ~<_  T )  ->  S  ~<_  T )
1613, 14, 15syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  ~<_  T )
17 numdom 8474 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  S  ~<_  T )  ->  S  e.  dom  card )
1811, 16, 17syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  S  e.  dom  card )
19 numdom 8474 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  dom  card  /\  R  ~<_  T )  ->  R  e.  dom  card )
2011, 14, 19syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  R  e.  dom  card )
21 cardsdom2 8427 . . . . . . . . . . . . . 14  |-  ( ( S  e.  dom  card  /\  R  e.  dom  card )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
2218, 20, 21syl2anc 667 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( card `  R )  e.  On  /\  ( card `  R )  C_  ( card `  T ) )  /\  R  ~<_  T )  /\  S  ~<  R )  ->  ( ( card `  S )  e.  (
card `  R )  <->  S 
~<  R ) )
239, 22mpbird 236 . . . . . . . . . . . 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 84 . . . . . . . . . . . 12  |-  ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ( ( (
card `  S )  e.  ( card `  R
)  ->  ( S  ~<_  T  ->  ch ) )  ->  ch ) ) )
2623, 16, 25sylc 62 . . . . . . . . . . 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 436 . . . . . . . . . 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 81 . . . . . . . . 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 1765 . . . . . . . 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 1208 . . . . . . 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 86 . . . . . 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 1223 . . . . 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 1269 . . . 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 436 . . 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 4415 . . . 4  |-  ( x  =  y  ->  ( R  ~<_  T  <->  S  ~<_  T ) )
38 indcardi.d . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
3937, 38imbi12d 322 . . 3  |-  ( x  =  y  ->  (
( R  ~<_  T  ->  ps )  <->  ( S  ~<_  T  ->  ch ) ) )
40 indcardi.g . . . . 5  |-  ( x  =  A  ->  R  =  T )
4140breq1d 4415 . . . 4  |-  ( x  =  A  ->  ( R  ~<_  T  <->  T  ~<_  T ) )
42 indcardi.e . . . 4  |-  ( x  =  A  ->  ( ps 
<->  th ) )
4341, 42imbi12d 322 . . 3  |-  ( x  =  A  ->  (
( R  ~<_  T  ->  ps )  <->  ( T  ~<_  T  ->  th ) ) )
4436fveq2d 5874 . . 3  |-  ( x  =  y  ->  ( card `  R )  =  ( card `  S
) )
4540fveq2d 5874 . . 3  |-  ( x  =  A  ->  ( card `  R )  =  ( card `  T
) )
464, 6, 35, 39, 43, 44, 45tfisi 6690 . 2  |-  ( ph  ->  ( T  ~<_  T  ->  th ) )
473, 46mpd 15 1  |-  ( ph  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986   A.wal 1444    = wceq 1446    e. wcel 1889    C_ wss 3406   class class class wbr 4405   dom cdm 4837   Oncon0 5426   ` cfv 5585    ~<_ cdom 7572    ~< csdm 7573   cardccrd 8374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-wrecs 7033  df-recs 7095  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-card 8378
This theorem is referenced by:  uzindi  12201  symggen  17123
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