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Theorem exmidfodomrlemeldju 7176
Description: Lemma for exmidfodomr 7181. A variant of djur 7046. (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 3147 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  1o )
3 el1o 6416 . . . . . . . . 9  |-  ( x  e.  1o  <->  x  =  (/) )
42, 3sylib 121 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  x  =  (/) )
54fveq2d 5500 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (inl `  x )  =  (inl
`  (/) ) )
65eqeq2d 2182 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  <->  B  =  (inl `  (/) ) ) )
76biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  (inl `  x
)  ->  B  =  (inl `  (/) ) ) )
87rexlimdva 2587 . . . 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 728 . 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 5500 . . . . . . 7  |-  ( (
ph  /\  x  e.  1o )  ->  (inr `  x )  =  (inr
`  (/) ) )
1413eqeq2d 2182 . . . . . 6  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  <->  B  =  (inr `  (/) ) ) )
1514biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  (inr `  x
)  ->  B  =  (inr `  (/) ) ) )
1615rexlimdva 2587 . . . 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 729 . 2  |-  ( (
ph  /\  E. x  e.  1o  B  =  (inr
`  x ) )  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
19 exmidfodomrlemeldju.el . . 3  |-  ( ph  ->  B  e.  ( A 1o ) )
20 djur 7046 . . 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 793 1  |-  ( ph  ->  ( B  =  (inl
`  (/) )  \/  B  =  (inr `  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348    e. wcel 2141   E.wrex 2449    C_ wss 3121   (/)c0 3414   ` cfv 5198   1oc1o 6388   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025
This theorem is referenced by:  exmidfodomrlemr  7179
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