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Theorem findcard2d 6919
Description: Deduction version of findcard2 6917. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6904), use findcard2sd 6920 instead. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
findcard2d.ch  |-  ( x  =  (/)  ->  ( ps  <->  ch ) )
findcard2d.th  |-  ( x  =  y  ->  ( ps 
<->  th ) )
findcard2d.ta  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ps  <->  ta )
)
findcard2d.et  |-  ( x  =  A  ->  ( ps 
<->  et ) )
findcard2d.z  |-  ( ph  ->  ch )
findcard2d.i  |-  ( (
ph  /\  ( y  C_  A  /\  z  e.  ( A  \  y
) ) )  -> 
( th  ->  ta ) )
findcard2d.a  |-  ( ph  ->  A  e.  Fin )
Assertion
Ref Expression
findcard2d  |-  ( ph  ->  et )
Distinct variable groups:    x, A, y, z    ph, x, y, z    ps, y, z    ch, x    th, x    ta, x    et, x
Allowed substitution hints:    ps( x)    ch( y,
z)    th( y, z)    ta( y, z)    et( y, z)

Proof of Theorem findcard2d
StepHypRef Expression
1 ssid 3190 . 2  |-  A  C_  A
2 findcard2d.a . . . 4  |-  ( ph  ->  A  e.  Fin )
32adantr 276 . . 3  |-  ( (
ph  /\  A  C_  A
)  ->  A  e.  Fin )
4 sseq1 3193 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  A  <->  (/)  C_  A
) )
54anbi2d 464 . . . . 5  |-  ( x  =  (/)  ->  ( (
ph  /\  x  C_  A
)  <->  ( ph  /\  (/)  C_  A ) ) )
6 findcard2d.ch . . . . 5  |-  ( x  =  (/)  ->  ( ps  <->  ch ) )
75, 6imbi12d 234 . . . 4  |-  ( x  =  (/)  ->  ( ( ( ph  /\  x  C_  A )  ->  ps ) 
<->  ( ( ph  /\  (/)  C_  A )  ->  ch ) ) )
8 sseq1 3193 . . . . . 6  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
98anbi2d 464 . . . . 5  |-  ( x  =  y  ->  (
( ph  /\  x  C_  A )  <->  ( ph  /\  y  C_  A )
) )
10 findcard2d.th . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  th ) )
119, 10imbi12d 234 . . . 4  |-  ( x  =  y  ->  (
( ( ph  /\  x  C_  A )  ->  ps )  <->  ( ( ph  /\  y  C_  A )  ->  th ) ) )
12 sseq1 3193 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( x  C_  A 
<->  ( y  u.  {
z } )  C_  A ) )
1312anbi2d 464 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( ph  /\  x  C_  A )  <->  (
ph  /\  ( y  u.  { z } ) 
C_  A ) ) )
14 findcard2d.ta . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ps  <->  ta )
)
1513, 14imbi12d 234 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( (
ph  /\  x  C_  A
)  ->  ps )  <->  ( ( ph  /\  (
y  u.  { z } )  C_  A
)  ->  ta )
) )
16 sseq1 3193 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
1716anbi2d 464 . . . . 5  |-  ( x  =  A  ->  (
( ph  /\  x  C_  A )  <->  ( ph  /\  A  C_  A )
) )
18 findcard2d.et . . . . 5  |-  ( x  =  A  ->  ( ps 
<->  et ) )
1917, 18imbi12d 234 . . . 4  |-  ( x  =  A  ->  (
( ( ph  /\  x  C_  A )  ->  ps )  <->  ( ( ph  /\  A  C_  A )  ->  et ) ) )
20 findcard2d.z . . . . 5  |-  ( ph  ->  ch )
2120adantr 276 . . . 4  |-  ( (
ph  /\  (/)  C_  A
)  ->  ch )
22 simprl 529 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  ->  ph )
23 simprr 531 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( y  u.  {
z } )  C_  A )
2423unssad 3327 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
y  C_  A )
2522, 24jca 306 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( ph  /\  y  C_  A ) )
26 id 19 . . . . . . . . . . 11  |-  ( ( y  u.  { z } )  C_  A  ->  ( y  u.  {
z } )  C_  A )
27 vsnid 3639 . . . . . . . . . . . 12  |-  z  e. 
{ z }
28 elun2 3318 . . . . . . . . . . . 12  |-  ( z  e.  { z }  ->  z  e.  ( y  u.  { z } ) )
2927, 28mp1i 10 . . . . . . . . . . 11  |-  ( ( y  u.  { z } )  C_  A  ->  z  e.  ( y  u.  { z } ) )
3026, 29sseldd 3171 . . . . . . . . . 10  |-  ( ( y  u.  { z } )  C_  A  ->  z  e.  A )
3130ad2antll 491 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
z  e.  A )
32 simplr 528 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  ->  -.  z  e.  y
)
3331, 32eldifd 3154 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
z  e.  ( A 
\  y ) )
34 findcard2d.i . . . . . . . 8  |-  ( (
ph  /\  ( y  C_  A  /\  z  e.  ( A  \  y
) ) )  -> 
( th  ->  ta ) )
3522, 24, 33, 34syl12anc 1247 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( th  ->  ta ) )
3625, 35embantd 56 . . . . . 6  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( ( ( ph  /\  y  C_  A )  ->  th )  ->  ta ) )
3736ex 115 . . . . 5  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( ( ph  /\  ( y  u. 
{ z } ) 
C_  A )  -> 
( ( ( ph  /\  y  C_  A )  ->  th )  ->  ta ) ) )
3837com23 78 . . . 4  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( (
( ph  /\  y  C_  A )  ->  th )  ->  ( ( ph  /\  ( y  u.  {
z } )  C_  A )  ->  ta ) ) )
397, 11, 15, 19, 21, 38findcard2s 6918 . . 3  |-  ( A  e.  Fin  ->  (
( ph  /\  A  C_  A )  ->  et ) )
403, 39mpcom 36 . 2  |-  ( (
ph  /\  A  C_  A
)  ->  et )
411, 40mpan2 425 1  |-  ( ph  ->  et )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    \ cdif 3141    u. cun 3142    C_ wss 3144   (/)c0 3437   {csn 3607   Fincfn 6766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6559  df-en 6767  df-fin 6769
This theorem is referenced by:  iunfidisj  6975
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