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Theorem cplem1 8310
Description: Lemma for the Collection Principle cp 8312. (Contributed by NM, 17-Oct-2003.)
Hypotheses
Ref Expression
cplem1.1  |-  C  =  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z
) }
cplem1.2  |-  D  = 
U_ x  e.  A  C
Assertion
Ref Expression
cplem1  |-  A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) )
Distinct variable groups:    x, y,
z, A    y, B, z
Allowed substitution hints:    B( x)    C( x, y, z)    D( x, y, z)

Proof of Theorem cplem1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 scott0 8307 . . . . . 6  |-  ( B  =  (/)  <->  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z
) }  =  (/) )
2 cplem1.1 . . . . . . 7  |-  C  =  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z
) }
32eqeq1i 2450 . . . . . 6  |-  ( C  =  (/)  <->  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z
) }  =  (/) )
41, 3bitr4i 252 . . . . 5  |-  ( B  =  (/)  <->  C  =  (/) )
54necon3bii 2711 . . . 4  |-  ( B  =/=  (/)  <->  C  =/=  (/) )
6 n0 3780 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
75, 6bitri 249 . . 3  |-  ( B  =/=  (/)  <->  E. w  w  e.  C )
8 ssrab2 3570 . . . . . . . . 9  |-  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z ) }  C_  B
92, 8eqsstri 3519 . . . . . . . 8  |-  C  C_  B
109sseli 3485 . . . . . . 7  |-  ( w  e.  C  ->  w  e.  B )
1110a1i 11 . . . . . 6  |-  ( x  e.  A  ->  (
w  e.  C  ->  w  e.  B )
)
12 ssiun2 4358 . . . . . . . 8  |-  ( x  e.  A  ->  C  C_ 
U_ x  e.  A  C )
13 cplem1.2 . . . . . . . 8  |-  D  = 
U_ x  e.  A  C
1412, 13syl6sseqr 3536 . . . . . . 7  |-  ( x  e.  A  ->  C  C_  D )
1514sseld 3488 . . . . . 6  |-  ( x  e.  A  ->  (
w  e.  C  ->  w  e.  D )
)
1611, 15jcad 533 . . . . 5  |-  ( x  e.  A  ->  (
w  e.  C  -> 
( w  e.  B  /\  w  e.  D
) ) )
17 inelcm 3867 . . . . 5  |-  ( ( w  e.  B  /\  w  e.  D )  ->  ( B  i^i  D
)  =/=  (/) )
1816, 17syl6 33 . . . 4  |-  ( x  e.  A  ->  (
w  e.  C  -> 
( B  i^i  D
)  =/=  (/) ) )
1918exlimdv 1711 . . 3  |-  ( x  e.  A  ->  ( E. w  w  e.  C  ->  ( B  i^i  D )  =/=  (/) ) )
207, 19syl5bi 217 . 2  |-  ( x  e.  A  ->  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) ) )
2120rgen 2803 1  |-  A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797    i^i cin 3460    C_ wss 3461   (/)c0 3770   U_ciun 4315   ` cfv 5578   rankcrnk 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186
This theorem is referenced by:  cplem2  8311
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