ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fodju0 Unicode version

Theorem fodju0 6987
Description: Lemma for fodjuomni 6989 and fodjumkv 7002. A condition which shows that  A is empty. (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 ) )
fodju0.1  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
Assertion
Ref Expression
fodju0  |-  ( ph  ->  A  =  (/) )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    y, A    y, F    w, O    w, P
Allowed substitution hints:    ph( w)    A( w)    B( y, w)    P( y,
z)    F( w)

Proof of Theorem fodju0
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuf.fo . . . . 5  |-  ( ph  ->  F : O -onto-> ( A B ) )
2 djulcl 6904 . . . . 5  |-  ( u  e.  A  ->  (inl `  u )  e.  ( A B ) )
3 foelrn 5622 . . . . 5  |-  ( ( F : O -onto-> ( A B )  /\  (inl `  u )  e.  ( A B ) )  ->  E. v  e.  O  (inl `  u )  =  ( F `  v
) )
41, 2, 3syl2an 287 . . . 4  |-  ( (
ph  /\  u  e.  A )  ->  E. v  e.  O  (inl `  u
)  =  ( F `
 v ) )
5 fodjuf.p . . . . . 6  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
6 fveqeq2 5398 . . . . . . . 8  |-  ( y  =  v  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  v )  =  (inl
`  z ) ) )
76rexbidv 2415 . . . . . . 7  |-  ( y  =  v  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  v )  =  (inl `  z
) ) )
87ifbid 3463 . . . . . 6  |-  ( y  =  v  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
9 simprl 505 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
v  e.  O )
10 peano1 4478 . . . . . . . 8  |-  (/)  e.  om
1110a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/) 
e.  om )
12 1onn 6384 . . . . . . . 8  |-  1o  e.  om
1312a1i 9 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  1o  e.  om )
141fodjuomnilemdc 6984 . . . . . . . 8  |-  ( (
ph  /\  v  e.  O )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
1514ad2ant2r 500 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> DECID  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
1611, 13, 15ifcldcd 3477 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
175, 8, 9, 16fvmptd3 5482 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  if ( E. z  e.  A  ( F `  v )  =  (inl `  z
) ,  (/) ,  1o ) )
18 fveqeq2 5398 . . . . . 6  |-  ( w  =  v  ->  (
( P `  w
)  =  1o  <->  ( P `  v )  =  1o ) )
19 fodju0.1 . . . . . . 7  |-  ( ph  ->  A. w  e.  O  ( P `  w )  =  1o )
2019ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  A. w  e.  O  ( P `  w )  =  1o )
2118, 20, 9rspcdva 2768 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( P `  v
)  =  1o )
22 simplr 504 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  u  e.  A )
23 simprr 506 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
(inl `  u )  =  ( F `  v ) )
2423eqcomd 2123 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  -> 
( F `  v
)  =  (inl `  u ) )
25 fveq2 5389 . . . . . . . 8  |-  ( z  =  u  ->  (inl `  z )  =  (inl
`  u ) )
2625rspceeqv 2781 . . . . . . 7  |-  ( ( u  e.  A  /\  ( F `  v )  =  (inl `  u
) )  ->  E. z  e.  A  ( F `  v )  =  (inl
`  z ) )
2722, 24, 26syl2anc 408 . . . . . 6  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  E. z  e.  A  ( F `  v )  =  (inl `  z
) )
2827iftrued 3451 . . . . 5  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  if ( E. z  e.  A  ( F `  v )  =  (inl
`  z ) ,  (/) ,  1o )  =  (/) )
2917, 21, 283eqtr3rd 2159 . . . 4  |-  ( ( ( ph  /\  u  e.  A )  /\  (
v  e.  O  /\  (inl `  u )  =  ( F `  v
) ) )  ->  (/)  =  1o )
304, 29rexlimddv 2531 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  (/)  =  1o )
31 1n0 6297 . . . . 5  |-  1o  =/=  (/)
3231nesymi 2331 . . . 4  |-  -.  (/)  =  1o
3332a1i 9 . . 3  |-  ( (
ph  /\  u  e.  A )  ->  -.  (/)  =  1o )
3430, 33pm2.65da 635 . 2  |-  ( ph  ->  -.  u  e.  A
)
3534eq0rdv 3377 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  DECID wdc 804    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   (/)c0 3333   ifcif 3444    |-> cmpt 3959   omcom 4474   -onto->wfo 5091   ` cfv 5093   1oc1o 6274   ⊔ cdju 6890  inlcinl 6898
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-13 1476  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  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  fodjuomnilemres  6988  fodjumkvlemres  7001
  Copyright terms: Public domain W3C validator