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Theorem cplem1 7773
Description: Lemma for the Collection Principle cp 7775. (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 7770 . . . . . 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 2415 . . . . . 6  |-  ( C  =  (/)  <->  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z
) }  =  (/) )
41, 3bitr4i 244 . . . . 5  |-  ( B  =  (/)  <->  C  =  (/) )
54necon3bii 2603 . . . 4  |-  ( B  =/=  (/)  <->  C  =/=  (/) )
6 n0 3601 . . . 4  |-  ( C  =/=  (/)  <->  E. w  w  e.  C )
75, 6bitri 241 . . 3  |-  ( B  =/=  (/)  <->  E. w  w  e.  C )
8 ssrab2 3392 . . . . . . . . 9  |-  { y  e.  B  |  A. z  e.  B  ( rank `  y )  C_  ( rank `  z ) }  C_  B
92, 8eqsstri 3342 . . . . . . . 8  |-  C  C_  B
109sseli 3308 . . . . . . 7  |-  ( w  e.  C  ->  w  e.  B )
1110a1i 11 . . . . . 6  |-  ( x  e.  A  ->  (
w  e.  C  ->  w  e.  B )
)
12 ssiun2 4098 . . . . . . . 8  |-  ( x  e.  A  ->  C  C_ 
U_ x  e.  A  C )
13 cplem1.2 . . . . . . . 8  |-  D  = 
U_ x  e.  A  C
1412, 13syl6sseqr 3359 . . . . . . 7  |-  ( x  e.  A  ->  C  C_  D )
1514sseld 3311 . . . . . 6  |-  ( x  e.  A  ->  (
w  e.  C  ->  w  e.  D )
)
1611, 15jcad 520 . . . . 5  |-  ( x  e.  A  ->  (
w  e.  C  -> 
( w  e.  B  /\  w  e.  D
) ) )
17 inelcm 3646 . . . . 5  |-  ( ( w  e.  B  /\  w  e.  D )  ->  ( B  i^i  D
)  =/=  (/) )
1816, 17syl6 31 . . . 4  |-  ( x  e.  A  ->  (
w  e.  C  -> 
( B  i^i  D
)  =/=  (/) ) )
1918exlimdv 1643 . . 3  |-  ( x  e.  A  ->  ( E. w  w  e.  C  ->  ( B  i^i  D )  =/=  (/) ) )
207, 19syl5bi 209 . 2  |-  ( x  e.  A  ->  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) ) )
2120rgen 2735 1  |-  A. x  e.  A  ( B  =/=  (/)  ->  ( B  i^i  D )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   {crab 2674    i^i cin 3283    C_ wss 3284   (/)c0 3592   U_ciun 4057   ` cfv 5417   rankcrnk 7649
This theorem is referenced by:  cplem2  7774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-r1 7650  df-rank 7651
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