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Theorem acexmidlem1 5539
Description: Lemma for acexmid 5542. List the cases identified in acexmidlemcase 5538 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 5538 . 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 5534 . . . 4  |-  ( {
(/) }  e.  A  ->  ph )
65orcd 685 . . 3  |-  ( {
(/) }  e.  A  ->  ( ph  \/  -.  ph ) )
71, 2, 3acexmidlemb 5535 . . . 4  |-  ( (/)  e.  B  ->  ph )
87orcd 685 . . 3  |-  ( (/)  e.  B  ->  ( ph  \/  -.  ph ) )
91, 2, 3acexmidlemab 5537 . . . 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 686 . . 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 1235 . 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 662    \/ w3o 919    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   E!wreu 2351   {crab 2353   (/)c0 3258   {csn 3406   {cpr 3407   iota_crio 5498
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134  df-iota 4897  df-riota 5499
This theorem is referenced by:  acexmidlem2  5540
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