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Theorem acexmidlem1 5738
Description: Lemma for acexmid 5741. List the cases identified in acexmidlemcase 5737 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 5737 . 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 5733 . . . 4  |-  ( {
(/) }  e.  A  ->  ph )
65orcd 707 . . 3  |-  ( {
(/) }  e.  A  ->  ( ph  \/  -.  ph ) )
71, 2, 3acexmidlemb 5734 . . . 4  |-  ( (/)  e.  B  ->  ph )
87orcd 707 . . 3  |-  ( (/)  e.  B  ->  ( ph  \/  -.  ph ) )
91, 2, 3acexmidlemab 5736 . . . 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 708 . . 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 1266 . 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 682    \/ w3o 946    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   E!wreu 2395   {crab 2397   (/)c0 3333   {csn 3497   {cpr 3498   iota_crio 5697
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iota 5058  df-riota 5698
This theorem is referenced by:  acexmidlem2  5739
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