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Theorem acexmidlem1 5773
Description: Lemma for acexmid 5776. List the cases identified in acexmidlemcase 5772 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 5772 . 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 5768 . . . 4  |-  ( {
(/) }  e.  A  ->  ph )
65orcd 722 . . 3  |-  ( {
(/) }  e.  A  ->  ( ph  \/  -.  ph ) )
71, 2, 3acexmidlemb 5769 . . . 4  |-  ( (/)  e.  B  ->  ph )
87orcd 722 . . 3  |-  ( (/)  e.  B  ->  ( ph  \/  -.  ph ) )
91, 2, 3acexmidlemab 5771 . . . 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 723 . . 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 1281 . 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 103    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   E!wreu 2418   {crab 2420   (/)c0 3363   {csn 3527   {cpr 3528   iota_crio 5732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-nul 4057  ax-pow 4101
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3740  df-tr 4030  df-iord 4291  df-on 4293  df-suc 4296  df-iota 5091  df-riota 5733
This theorem is referenced by:  acexmidlem2  5774
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