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Theorem fodjum 7024
Description: Lemma for fodjuomni 7027 and fodjumkv 7040. A condition which shows that  A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjuf.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
fodjuf.p  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
fodjum.z  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
Assertion
Ref Expression
fodjum  |-  ( ph  ->  E. x  x  e.  A )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    w, A, x, z    y, A, w    y, F    ph, w
Allowed substitution hints:    ph( x)    B( x, y, w)    P( x, y, z, w)    F( x, w)    O( x, w)

Proof of Theorem fodjum
StepHypRef Expression
1 fodjum.z . 2  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
2 1n0 6335 . . . . . . . . 9  |-  1o  =/=  (/)
32nesymi 2355 . . . . . . . 8  |-  -.  (/)  =  1o
43intnan 915 . . . . . . 7  |-  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o )
54a1i 9 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  1o ) )
6 simprr 522 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  (/) )
7 fodjuf.p . . . . . . . . 9  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
8 fveqeq2 5436 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  w )  =  (inl
`  z ) ) )
98rexbidv 2439 . . . . . . . . . 10  |-  ( y  =  w  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  w )  =  (inl `  z
) ) )
109ifbid 3496 . . . . . . . . 9  |-  ( y  =  w  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
11 simprl 521 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  w  e.  O )
12 peano1 4514 . . . . . . . . . . 11  |-  (/)  e.  om
1312a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/) 
e.  om )
14 1onn 6422 . . . . . . . . . . 11  |-  1o  e.  om
1514a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  1o  e.  om )
16 fodjuf.fo . . . . . . . . . . . 12  |-  ( ph  ->  F : O -onto-> ( A B ) )
1716fodjuomnilemdc 7022 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  O )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
1817adantrr 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z ) )
1913, 15, 18ifcldcd 3510 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  if ( E. z  e.  A  ( F `  w )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
207, 10, 11, 19fvmptd3 5520 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
216, 20eqtr3d 2175 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
22 eqifdc 3509 . . . . . . . 8  |-  (DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2318, 22syl 14 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2421, 23mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) )
255, 24ecased 1328 . . . . 5  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) ) )
2625simpld 111 . . . 4  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
27 rexm 3465 . . . 4  |-  ( E. z  e.  A  ( F `  w )  =  (inl `  z
)  ->  E. z 
z  e.  A )
2826, 27syl 14 . . 3  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  z  e.  A )
29 eleq1w 2201 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
3029cbvexv 1891 . . 3  |-  ( E. z  z  e.  A  <->  E. x  x  e.  A
)
3128, 30sylib 121 . 2  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. x  x  e.  A )
321, 31rexlimddv 2557 1  |-  ( ph  ->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1332   E.wex 1469    e. wcel 1481   E.wrex 2418   (/)c0 3366   ifcif 3477    |-> cmpt 3995   omcom 4510   -onto->wfo 5127   ` cfv 5129   1oc1o 6312   ⊔ cdju 6928  inlcinl 6936
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-iord 4294  df-on 4296  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-1st 6044  df-2nd 6045  df-1o 6319  df-dju 6929  df-inl 6938  df-inr 6939
This theorem is referenced by:  fodjuomnilemres  7026  fodjumkvlemres  7039
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