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Theorem tz9.1 8163
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8162 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 2443 . . 3  |-  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )  =  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )
3 eqid 2443 . . 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 8162 . 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 8063 . . . 4  |-  om  e.  _V
6 fvex 5866 . . . 4  |-  ( ( rec ( ( w  e.  _V  |->  ( w  u.  U. w ) ) ,  A )  |`  om ) `  z
)  e.  _V
75, 6iunex 6765 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  e.  _V
8 sseq2 3511 . . . 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 4536 . . . 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 3510 . . . . . 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 316 . . . . 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 1700 . . . 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 1300 . . 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 3187 . 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 5 1  |-  E. x
( A  C_  x  /\  Tr  x  /\  A. y ( ( A 
C_  y  /\  Tr  y )  ->  x  C_  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    u. cun 3459    C_ wss 3461   U.cuni 4234   U_ciun 4315    |-> cmpt 4495   Tr wtr 4530    |` cres 4991   ` cfv 5578   omcom 6685   reccrdg 7077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078
This theorem is referenced by:  epfrs  8165
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