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Theorem findcard2d 6379
Description: Deduction version of findcard2 6377. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6367), use findcard2sd 6380 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 2992 . 2  |-  A  C_  A
2 findcard2d.a . . . 4  |-  ( ph  ->  A  e.  Fin )
32adantr 265 . . 3  |-  ( (
ph  /\  A  C_  A
)  ->  A  e.  Fin )
4 sseq1 2994 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  A  <->  (/)  C_  A
) )
54anbi2d 445 . . . . 5  |-  ( x  =  (/)  ->  ( (
ph  /\  x  C_  A
)  <->  ( ph  /\  (/)  C_  A ) ) )
6 findcard2d.ch . . . . 5  |-  ( x  =  (/)  ->  ( ps  <->  ch ) )
75, 6imbi12d 227 . . . 4  |-  ( x  =  (/)  ->  ( ( ( ph  /\  x  C_  A )  ->  ps ) 
<->  ( ( ph  /\  (/)  C_  A )  ->  ch ) ) )
8 sseq1 2994 . . . . . 6  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
98anbi2d 445 . . . . 5  |-  ( x  =  y  ->  (
( ph  /\  x  C_  A )  <->  ( ph  /\  y  C_  A )
) )
10 findcard2d.th . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  th ) )
119, 10imbi12d 227 . . . 4  |-  ( x  =  y  ->  (
( ( ph  /\  x  C_  A )  ->  ps )  <->  ( ( ph  /\  y  C_  A )  ->  th ) ) )
12 sseq1 2994 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( x  C_  A 
<->  ( y  u.  {
z } )  C_  A ) )
1312anbi2d 445 . . . . 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 227 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( (
ph  /\  x  C_  A
)  ->  ps )  <->  ( ( ph  /\  (
y  u.  { z } )  C_  A
)  ->  ta )
) )
16 sseq1 2994 . . . . . 6  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
1716anbi2d 445 . . . . 5  |-  ( x  =  A  ->  (
( ph  /\  x  C_  A )  <->  ( ph  /\  A  C_  A )
) )
18 findcard2d.et . . . . 5  |-  ( x  =  A  ->  ( ps 
<->  et ) )
1917, 18imbi12d 227 . . . 4  |-  ( x  =  A  ->  (
( ( ph  /\  x  C_  A )  ->  ps )  <->  ( ( ph  /\  A  C_  A )  ->  et ) ) )
20 findcard2d.z . . . . 5  |-  ( ph  ->  ch )
2120adantr 265 . . . 4  |-  ( (
ph  /\  (/)  C_  A
)  ->  ch )
22 simprl 491 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  ->  ph )
23 simprr 492 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( y  u.  {
z } )  C_  A )
2423unssad 3148 . . . . . . . 8  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
y  C_  A )
2522, 24jca 294 . . . . . . 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 3431 . . . . . . . . . . . 12  |-  z  e. 
{ z }
28 elun2 3139 . . . . . . . . . . . 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 2974 . . . . . . . . . 10  |-  ( ( y  u.  { z } )  C_  A  ->  z  e.  A )
3130ad2antll 468 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
z  e.  A )
32 simplr 490 . . . . . . . . 9  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  ->  -.  z  e.  y
)
3331, 32eldifd 2956 . . . . . . . 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 1144 . . . . . . 7  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( th  ->  ta ) )
3625, 35embantd 54 . . . . . 6  |-  ( ( ( y  e.  Fin  /\ 
-.  z  e.  y )  /\  ( ph  /\  ( y  u.  {
z } )  C_  A ) )  -> 
( ( ( ph  /\  y  C_  A )  ->  th )  ->  ta ) )
3736ex 112 . . . . 5  |-  ( ( y  e.  Fin  /\  -.  z  e.  y
)  ->  ( ( ph  /\  ( y  u. 
{ z } ) 
C_  A )  -> 
( ( ( ph  /\  y  C_  A )  ->  th )  ->  ta ) ) )
3837com23 76 . . . 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 6378 . . 3  |-  ( A  e.  Fin  ->  (
( ph  /\  A  C_  A )  ->  et ) )
403, 39mpcom 36 . 2  |-  ( (
ph  /\  A  C_  A
)  ->  et )
411, 40mpan2 409 1  |-  ( ph  ->  et )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409    \ cdif 2942    u. cun 2943    C_ wss 2945   (/)c0 3252   {csn 3403   Fincfn 6252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-er 6137  df-en 6253  df-fin 6255
This theorem is referenced by: (None)
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