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Theorem scott0f 32424
Description: A version of scott0 8362 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
scott0f.1  |-  F/_ y A
scott0f.2  |-  F/_ x A
Assertion
Ref Expression
scott0f  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem scott0f
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scott0 8362 . 2  |-  ( A  =  (/)  <->  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z
) }  =  (/) )
2 scott0f.1 . . . . . . 7  |-  F/_ y A
3 nfcv 2594 . . . . . . 7  |-  F/_ z A
4 nfv 1763 . . . . . . 7  |-  F/ z ( rank `  x
)  C_  ( rank `  y )
5 nfv 1763 . . . . . . 7  |-  F/ y ( rank `  x
)  C_  ( rank `  z )
6 fveq2 5870 . . . . . . . 8  |-  ( y  =  z  ->  ( rank `  y )  =  ( rank `  z
) )
76sseq2d 3462 . . . . . . 7  |-  ( y  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
82, 3, 4, 5, 7cbvralf 3015 . . . . . 6  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  <->  A. z  e.  A  (
rank `  x )  C_  ( rank `  z
) )
98a1i 11 . . . . 5  |-  ( x  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  <->  A. z  e.  A  ( rank `  x )  C_  ( rank `  z
) ) )
109rabbiia 3035 . . . 4  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  { x  e.  A  |  A. z  e.  A  ( rank `  x )  C_  ( rank `  z ) }
11 nfcv 2594 . . . . 5  |-  F/_ w A
12 scott0f.2 . . . . 5  |-  F/_ x A
13 nfv 1763 . . . . . 6  |-  F/ x
( rank `  w )  C_  ( rank `  z
)
1412, 13nfral 2776 . . . . 5  |-  F/ x A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)
15 nfv 1763 . . . . 5  |-  F/ w A. z  e.  A  ( rank `  x )  C_  ( rank `  z
)
16 fveq2 5870 . . . . . . 7  |-  ( w  =  x  ->  ( rank `  w )  =  ( rank `  x
) )
1716sseq1d 3461 . . . . . 6  |-  ( w  =  x  ->  (
( rank `  w )  C_  ( rank `  z
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
1817ralbidv 2829 . . . . 5  |-  ( w  =  x  ->  ( A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)  <->  A. z  e.  A  ( rank `  x )  C_  ( rank `  z
) ) )
1911, 12, 14, 15, 18cbvrab 3045 . . . 4  |-  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }  =  { x  e.  A  |  A. z  e.  A  ( rank `  x )  C_  ( rank `  z ) }
2010, 19eqtr4i 2478 . . 3  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }
2120eqeq1i 2458 . 2  |-  ( { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/)  <->  {
w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z
) }  =  (/) )
221, 21bitr4i 256 1  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1446    e. wcel 1889   F/_wnfc 2581   A.wral 2739   {crab 2743    C_ wss 3406   (/)c0 3733   ` cfv 5585   rankcrnk 8239
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-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-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-iin 4284  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-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-lim 5431  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-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-r1 8240  df-rank 8241
This theorem is referenced by:  scottn0f  32425
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