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Theorem scott0f 30825
Description: A version of scott0 8321 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 8321 . 2  |-  ( A  =  (/)  <->  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z
) }  =  (/) )
2 scott0f.1 . . . . . . 7  |-  F/_ y A
3 nfcv 2619 . . . . . . 7  |-  F/_ z A
4 nfv 1708 . . . . . . 7  |-  F/ z ( rank `  x
)  C_  ( rank `  y )
5 nfv 1708 . . . . . . 7  |-  F/ y ( rank `  x
)  C_  ( rank `  z )
6 fveq2 5872 . . . . . . . 8  |-  ( y  =  z  ->  ( rank `  y )  =  ( rank `  z
) )
76sseq2d 3527 . . . . . . 7  |-  ( y  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
82, 3, 4, 5, 7cbvralf 3078 . . . . . 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 3098 . . . 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 2619 . . . . 5  |-  F/_ w A
12 scott0f.2 . . . . 5  |-  F/_ x A
13 nfv 1708 . . . . . 6  |-  F/ x
( rank `  w )  C_  ( rank `  z
)
1412, 13nfral 2843 . . . . 5  |-  F/ x A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)
15 nfv 1708 . . . . 5  |-  F/ w A. z  e.  A  ( rank `  x )  C_  ( rank `  z
)
16 fveq2 5872 . . . . . . 7  |-  ( w  =  x  ->  ( rank `  w )  =  ( rank `  x
) )
1716sseq1d 3526 . . . . . 6  |-  ( w  =  x  ->  (
( rank `  w )  C_  ( rank `  z
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
1817ralbidv 2896 . . . . 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 3107 . . . 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 2489 . . 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 2464 . 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 252 1  |-  ( A  =  (/)  <->  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y
) }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   F/_wnfc 2605   A.wral 2807   {crab 2811    C_ wss 3471   (/)c0 3793   ` cfv 5594   rankcrnk 8198
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-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-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-iin 4335  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-we 4849  df-ord 4890  df-on 4891  df-lim 4892  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-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200
This theorem is referenced by:  scottn0f  30826
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