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Theorem exmidfodomrlemeldju 6825
Description: Lemma for exmidfodomr 6830. A variant of djur 6757. (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 3025 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  1o )
3 el1o 6201 . . . . . . . . 9  |-  ( x  e.  1o  <->  x  =  (/) )
42, 3sylib 120 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  x  =  (/) )
54fveq2d 5309 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (inl `  x )  =  (inl
`  (/) ) )
65eqeq2d 2099 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  <->  B  =  (inl `  (/) ) ) )
76biimpd 142 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  ->  B  =  (inl `  (/) ) ) )
87rexlimdva 2489 . . . 4  |-  ( ph  ->  ( E. x  e.  A  B  =  (inl
`  x )  ->  B  =  (inl `  (/) ) ) )
98imp 122 . . 3  |-  ( (
ph  /\  E. x  e.  A  B  =  (inl `  x ) )  ->  B  =  (inl
`  (/) ) )
109orcd 687 . 2  |-  ( (
ph  /\  E. x  e.  A  B  =  (inl `  x ) )  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
11 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  1o )  ->  x  e.  1o )
1211, 3sylib 120 . . . . . . . 8  |-  ( (
ph  /\  x  e.  1o )  ->  x  =  (/) )
1312fveq2d 5309 . . . . . . 7  |-  ( (
ph  /\  x  e.  1o )  ->  (inr `  x )  =  (inr
`  (/) ) )
1413eqeq2d 2099 . . . . . 6  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  <->  B  =  (inr `  (/) ) ) )
1514biimpd 142 . . . . 5  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  ->  B  =  (inr `  (/) ) ) )
1615rexlimdva 2489 . . . 4  |-  ( ph  ->  ( E. x  e.  1o  B  =  (inr
`  x )  ->  B  =  (inr `  (/) ) ) )
1716imp 122 . . 3  |-  ( (
ph  /\  E. x  e.  1o  B  =  (inr
`  x ) )  ->  B  =  (inr
`  (/) ) )
1817olcd 688 . 2  |-  ( (
ph  /\  E. x  e.  1o  B  =  (inr
`  x ) )  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
19 exmidfodomrlemeldju.el . . 3  |-  ( ph  ->  B  e.  ( A 1o ) )
20 djur 6757 . . 3  |-  ( B  e.  ( A 1o )  ->  ( E. x  e.  A  B  =  (inl `  x )  \/ 
E. x  e.  1o  B  =  (inr `  x
) ) )
2119, 20syl 14 . 2  |-  ( ph  ->  ( E. x  e.  A  B  =  (inl
`  x )  \/ 
E. x  e.  1o  B  =  (inr `  x
) ) )
2210, 18, 21mpjaodan 747 1  |-  ( ph  ->  ( B  =  (inl
`  (/) )  \/  B  =  (inr `  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   (/)c0 3286   ` cfv 5015   1oc1o 6174   ⊔ cdju 6730  inlcinl 6737  inrcinr 6738
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6731  df-inl 6739  df-inr 6740
This theorem is referenced by:  exmidfodomrlemr  6828
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