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Theorem acexmidlem1 5640
Description: Lemma for acexmid 5643. List the cases identified in acexmidlemcase 5639 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
acexmidlem.b  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
acexmidlem.c  |-  C  =  { A ,  B }
Assertion
Ref Expression
acexmidlem1  |-  ( A. z  e.  C  E! v  e.  z  E. u  e.  y  (
z  e.  u  /\  v  e.  u )  ->  ( ph  \/  -.  ph ) )
Distinct variable groups:    x, y, z, v, u, A    x, B, y, z, v, u   
x, C, y, z, v, u    ph, x, y, z, v, u

Proof of Theorem acexmidlem1
StepHypRef Expression
1 acexmidlem.a . . 3  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
2 acexmidlem.b . . 3  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
3 acexmidlem.c . . 3  |-  C  =  { A ,  B }
41, 2, 3acexmidlemcase 5639 . 2  |-  ( A. z  e.  C  E! v  e.  z  E. u  e.  y  (
z  e.  u  /\  v  e.  u )  ->  ( { (/) }  e.  A  \/  (/)  e.  B  \/  ( ( iota_ v  e.  A  E. u  e.  y  ( A  e.  u  /\  v  e.  u ) )  =  (/)  /\  ( iota_ v  e.  B  E. u  e.  y  ( B  e.  u  /\  v  e.  u ) )  =  { (/) } ) ) )
51, 2, 3acexmidlema 5635 . . . 4  |-  ( {
(/) }  e.  A  ->  ph )
65orcd 687 . . 3  |-  ( {
(/) }  e.  A  ->  ( ph  \/  -.  ph ) )
71, 2, 3acexmidlemb 5636 . . . 4  |-  ( (/)  e.  B  ->  ph )
87orcd 687 . . 3  |-  ( (/)  e.  B  ->  ( ph  \/  -.  ph ) )
91, 2, 3acexmidlemab 5638 . . . 4  |-  ( ( ( iota_ v  e.  A  E. u  e.  y 
( A  e.  u  /\  v  e.  u
) )  =  (/)  /\  ( iota_ v  e.  B  E. u  e.  y 
( B  e.  u  /\  v  e.  u
) )  =  { (/)
} )  ->  -.  ph )
109olcd 688 . . 3  |-  ( ( ( iota_ v  e.  A  E. u  e.  y 
( A  e.  u  /\  v  e.  u
) )  =  (/)  /\  ( iota_ v  e.  B  E. u  e.  y 
( B  e.  u  /\  v  e.  u
) )  =  { (/)
} )  ->  ( ph  \/  -.  ph )
)
116, 8, 103jaoi 1239 . 2  |-  ( ( { (/) }  e.  A  \/  (/)  e.  B  \/  ( ( iota_ v  e.  A  E. u  e.  y  ( A  e.  u  /\  v  e.  u ) )  =  (/)  /\  ( iota_ v  e.  B  E. u  e.  y  ( B  e.  u  /\  v  e.  u ) )  =  { (/) } ) )  ->  ( ph  \/  -.  ph ) )
124, 11syl 14 1  |-  ( A. z  e.  C  E! v  e.  z  E. u  e.  y  (
z  e.  u  /\  v  e.  u )  ->  ( ph  \/  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   E!wreu 2361   {crab 2363   (/)c0 3286   {csn 3444   {cpr 3445   iota_crio 5599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-iord 4191  df-on 4193  df-suc 4196  df-iota 4975  df-riota 5600
This theorem is referenced by:  acexmidlem2  5641
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