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Theorem tz9.1 7379
Description: Every set has a transitive closure (smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7378 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself?

(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)

Hypothesis
Ref Expression
tz9.1.1  |-  A  e. 
_V
Assertion
Ref Expression
tz9.1  |-  E. x
( A  C_  x  /\  Tr  x  /\  A. y ( ( A 
C_  y  /\  Tr  y )  ->  x  C_  y ) )
Distinct variable group:    x, A, y

Proof of Theorem tz9.1
StepHypRef Expression
1 tz9.1.1 . . 3  |-  A  e. 
_V
2 eqid 2258 . . 3  |-  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )  =  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )
3 eqid 2258 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )
41, 2, 3trcl 7378 . 2  |-  ( A 
C_  U_ z  e.  om  ( ( rec (
( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z )  /\  Tr  U_ z  e.  om  (
( rec ( ( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z )  /\  A. y ( ( A 
C_  y  /\  Tr  y )  ->  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z ) 
C_  y ) )
5 omex 7312 . . . 4  |-  om  e.  _V
6 fvex 5472 . . . 4  |-  ( ( rec ( ( w  e.  _V  |->  ( w  u.  U. w ) ) ,  A )  |`  om ) `  z
)  e.  _V
75, 6iunex 5704 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  e.  _V
8 sseq2 3175 . . . 4  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( A  C_  x 
<->  A  C_  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z ) ) )
9 treq 4093 . . . 4  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( Tr  x  <->  Tr 
U_ z  e.  om  ( ( rec (
( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z ) ) )
10 sseq1 3174 . . . . . 6  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( x  C_  y 
<-> 
U_ z  e.  om  ( ( rec (
( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z )  C_  y
) )
1110imbi2d 309 . . . . 5  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( ( ( A  C_  y  /\  Tr  y )  ->  x  C_  y )  <->  ( ( A  C_  y  /\  Tr  y )  ->  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z ) 
C_  y ) ) )
1211albidv 2005 . . . 4  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( A. y
( ( A  C_  y  /\  Tr  y )  ->  x  C_  y
)  <->  A. y ( ( A  C_  y  /\  Tr  y )  ->  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z ) 
C_  y ) ) )
138, 9, 123anbi123d 1257 . . 3  |-  ( x  =  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  ->  ( ( A 
C_  x  /\  Tr  x  /\  A. y ( ( A  C_  y  /\  Tr  y )  ->  x  C_  y ) )  <-> 
( A  C_  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  /\  Tr  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  /\  A. y ( ( A  C_  y  /\  Tr  y )  ->  U_ z  e.  om  ( ( rec (
( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z )  C_  y
) ) ) )
147, 13cla4ev 2850 . 2  |-  ( ( A  C_  U_ z  e. 
om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  /\  Tr  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  /\  A. y ( ( A  C_  y  /\  Tr  y )  ->  U_ z  e.  om  ( ( rec (
( w  e.  _V  |->  ( w  u.  U. w
) ) ,  A
)  |`  om ) `  z )  C_  y
) )  ->  E. x
( A  C_  x  /\  Tr  x  /\  A. y ( ( A 
C_  y  /\  Tr  y )  ->  x  C_  y ) ) )
154, 14ax-mp 10 1  |-  E. x
( A  C_  x  /\  Tr  x  /\  A. y ( ( A 
C_  y  /\  Tr  y )  ->  x  C_  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2763    u. cun 3125    C_ wss 3127   U.cuni 3801   U_ciun 3879    e. cmpt 4051   Tr wtr 4087   omcom 4628    |` cres 4663   ` cfv 4673   reccrdg 6390
This theorem is referenced by:  epfrs  7381
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484  ax-inf2 7310
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391
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