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Theorem exmidfodomrlemeldju 7136
Description: Lemma for exmidfodomr 7141. A variant of djur 7015. (Contributed by Jim Kingdon, 2-Jul-2022.)
Hypotheses
Ref Expression
exmidfodomrlemeldju.a  |-  ( ph  ->  A  C_  1o )
exmidfodomrlemeldju.el  |-  ( ph  ->  B  e.  ( A 1o ) )
Assertion
Ref Expression
exmidfodomrlemeldju  |-  ( ph  ->  ( B  =  (inl
`  (/) )  \/  B  =  (inr `  (/) ) ) )

Proof of Theorem exmidfodomrlemeldju
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 exmidfodomrlemeldju.a . . . . . . . . . 10  |-  ( ph  ->  A  C_  1o )
21sselda 3128 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  1o )
3 el1o 6386 . . . . . . . . 9  |-  ( x  e.  1o  <->  x  =  (/) )
42, 3sylib 121 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  x  =  (/) )
54fveq2d 5474 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (inl `  x )  =  (inl
`  (/) ) )
65eqeq2d 2169 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  <->  B  =  (inl `  (/) ) ) )
76biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  ->  B  =  (inl `  (/) ) ) )
87rexlimdva 2574 . . . 4  |-  ( ph  ->  ( E. x  e.  A  B  =  (inl
`  x )  ->  B  =  (inl `  (/) ) ) )
98imp 123 . . 3  |-  ( (
ph  /\  E. x  e.  A  B  =  (inl `  x ) )  ->  B  =  (inl
`  (/) ) )
109orcd 723 . 2  |-  ( (
ph  /\  E. x  e.  A  B  =  (inl `  x ) )  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
11 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  1o )  ->  x  e.  1o )
1211, 3sylib 121 . . . . . . . 8  |-  ( (
ph  /\  x  e.  1o )  ->  x  =  (/) )
1312fveq2d 5474 . . . . . . 7  |-  ( (
ph  /\  x  e.  1o )  ->  (inr `  x )  =  (inr
`  (/) ) )
1413eqeq2d 2169 . . . . . 6  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  <->  B  =  (inr `  (/) ) ) )
1514biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  ->  B  =  (inr `  (/) ) ) )
1615rexlimdva 2574 . . . 4  |-  ( ph  ->  ( E. x  e.  1o  B  =  (inr
`  x )  ->  B  =  (inr `  (/) ) ) )
1716imp 123 . . 3  |-  ( (
ph  /\  E. x  e.  1o  B  =  (inr
`  x ) )  ->  B  =  (inr
`  (/) ) )
1817olcd 724 . 2  |-  ( (
ph  /\  E. x  e.  1o  B  =  (inr
`  x ) )  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
19 exmidfodomrlemeldju.el . . 3  |-  ( ph  ->  B  e.  ( A 1o ) )
20 djur 7015 . . 3  |-  ( B  e.  ( A 1o )  <-> 
( E. x  e.  A  B  =  (inl
`  x )  \/ 
E. x  e.  1o  B  =  (inr `  x
) ) )
2119, 20sylib 121 . 2  |-  ( ph  ->  ( E. x  e.  A  B  =  (inl
`  x )  \/ 
E. x  e.  1o  B  =  (inr `  x
) ) )
2210, 18, 21mpjaodan 788 1  |-  ( ph  ->  ( B  =  (inl
`  (/) )  \/  B  =  (inr `  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1335    e. wcel 2128   E.wrex 2436    C_ wss 3102   (/)c0 3395   ` cfv 5172   1oc1o 6358   ⊔ cdju 6983  inlcinl 6991  inrcinr 6992
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-1st 6090  df-2nd 6091  df-1o 6365  df-dju 6984  df-inl 6993  df-inr 6994
This theorem is referenced by:  exmidfodomrlemr  7139
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