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Theorem tz9.1 7407
Description: Every set has a transitive closure (smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7406 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
Dummy variables  z  w are mutually distinct and distinct from all other variables.

Proof of Theorem tz9.1
StepHypRef Expression
1 tz9.1.1 . . 3  |-  A  e. 
_V
2 eqid 2285 . . 3  |-  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )  =  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )
3 eqid 2285 . . 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 7406 . 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 7340 . . . 4  |-  om  e.  _V
6 fvex 5500 . . . 4  |-  ( ( rec ( ( w  e.  _V  |->  ( w  u.  U. w ) ) ,  A )  |`  om ) `  z
)  e.  _V
75, 6iunex 5732 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  e.  _V
8 sseq2 3202 . . . 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 4121 . . . 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 3201 . . . . . 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 1612 . . . 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 1254 . . 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 2877 . 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 936   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   _Vcvv 2790    u. cun 3152    C_ wss 3154   U.cuni 3829   U_ciun 3907    e. cmpt 4079   Tr wtr 4115   omcom 4656    |` cres 4691   ` cfv 5222   reccrdg 6418
This theorem is referenced by:  epfrs  7409
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384  df-rdg 6419
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