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Theorem exmidmotap 7260
Description: The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
Assertion
Ref Expression
exmidmotap  |-  (EXMID  <->  A. x E* r  r TAp  x
)
Distinct variable group:    x, r

Proof of Theorem exmidmotap
Dummy variables  s  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . . . . . . . 8  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
r TAp  x )
2 exmidapne 7259 . . . . . . . . 9  |-  (EXMID  ->  (
r TAp  x  <->  r  =  { <. u ,  v
>.  |  ( (
u  e.  x  /\  v  e.  x )  /\  u  =/=  v
) } ) )
32adantr 276 . . . . . . . 8  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
( r TAp  x  <->  r  =  { <. u ,  v
>.  |  ( (
u  e.  x  /\  v  e.  x )  /\  u  =/=  v
) } ) )
41, 3mpbid 147 . . . . . . 7  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
r  =  { <. u ,  v >.  |  ( ( u  e.  x  /\  v  e.  x
)  /\  u  =/=  v ) } )
5 simprr 531 . . . . . . . 8  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
s TAp  x )
6 exmidapne 7259 . . . . . . . . 9  |-  (EXMID  ->  (
s TAp  x  <->  s  =  { <. u ,  v
>.  |  ( (
u  e.  x  /\  v  e.  x )  /\  u  =/=  v
) } ) )
76adantr 276 . . . . . . . 8  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
( s TAp  x  <->  s  =  { <. u ,  v
>.  |  ( (
u  e.  x  /\  v  e.  x )  /\  u  =/=  v
) } ) )
85, 7mpbid 147 . . . . . . 7  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
s  =  { <. u ,  v >.  |  ( ( u  e.  x  /\  v  e.  x
)  /\  u  =/=  v ) } )
94, 8eqtr4d 2213 . . . . . 6  |-  ( (EXMID  /\  ( r TAp  x  /\  s TAp  x ) )  -> 
r  =  s )
109ex 115 . . . . 5  |-  (EXMID  ->  (
( r TAp  x  /\  s TAp  x )  ->  r  =  s ) )
1110alrimivv 1875 . . . 4  |-  (EXMID  ->  A. r A. s ( ( r TAp  x  /\  s TAp  x
)  ->  r  =  s ) )
12 tapeq1 7251 . . . . 5  |-  ( r  =  s  ->  (
r TAp  x  <->  s TAp  x
) )
1312mo4 2087 . . . 4  |-  ( E* r  r TAp  x  <->  A. r A. s ( ( r TAp  x  /\  s TAp  x
)  ->  r  =  s ) )
1411, 13sylibr 134 . . 3  |-  (EXMID  ->  E* r  r TAp  x )
1514alrimiv 1874 . 2  |-  (EXMID  ->  A. x E* r  r TAp  x
)
16 2onn 6522 . . . 4  |-  2o  e.  om
17 tapeq2 7252 . . . . . 6  |-  ( x  =  2o  ->  (
r TAp  x  <->  r TAp  2o ) )
1817mobidv 2062 . . . . 5  |-  ( x  =  2o  ->  ( E* r  r TAp  x  <->  E* r  r TAp  2o ) )
1918spcgv 2825 . . . 4  |-  ( 2o  e.  om  ->  ( A. x E* r  r TAp  x  ->  E* r 
r TAp  2o ) )
2016, 19ax-mp 5 . . 3  |-  ( A. x E* r  r TAp  x  ->  E* r  r TAp  2o )
21 2omotap 7258 . . 3  |-  ( E* r  r TAp  2o  -> EXMID )
2220, 21syl 14 . 2  |-  ( A. x E* r  r TAp  x  -> EXMID )
2315, 22impbii 126 1  |-  (EXMID  <->  A. x E* r  r TAp  x
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    = wceq 1353   E*wmo 2027    e. wcel 2148    =/= wne 2347   {copab 4064  EXMIDwem 4195   omcom 4590   2oc2o 6411   TAp wtap 7248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-exmid 4196  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fo 5223  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-2o 6418  df-pap 7247  df-tap 7249
This theorem is referenced by: (None)
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