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Theorem tz9.1 7427
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7426 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
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.1.1 . . 3  |-  A  e. 
_V
2 eqid 2296 . . 3  |-  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )  =  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )
3 eqid 2296 . . 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 7426 . 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 7360 . . . 4  |-  om  e.  _V
6 fvex 5555 . . . 4  |-  ( ( rec ( ( w  e.  _V  |->  ( w  u.  U. w ) ) ,  A )  |`  om ) `  z
)  e.  _V
75, 6iunex 5786 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  e.  _V
8 sseq2 3213 . . . 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 4135 . . . 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 3212 . . . . . 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 307 . . . . 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 1615 . . . 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 1252 . . 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, 13spcev 2888 . 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 8 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 4    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   U.cuni 3843   U_ciun 3921    e. cmpt 4093   Tr wtr 4129   omcom 4672    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  epfrs  7429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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