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Theorem tz9.1 8064
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8063 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 2454 . . 3  |-  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )  =  ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om )
3 eqid 2454 . . 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 8063 . 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 7964 . . . 4  |-  om  e.  _V
6 fvex 5812 . . . 4  |-  ( ( rec ( ( w  e.  _V  |->  ( w  u.  U. w ) ) ,  A )  |`  om ) `  z
)  e.  _V
75, 6iunex 6670 . . 3  |-  U_ z  e.  om  ( ( rec ( ( w  e. 
_V  |->  ( w  u. 
U. w ) ) ,  A )  |`  om ) `  z )  e.  _V
8 sseq2 3489 . . . 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 4502 . . . 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 3488 . . . . . 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 1680 . . . 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 1290 . . 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 3170 . 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 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3078    u. cun 3437    C_ wss 3439   U.cuni 4202   U_ciun 4282    |-> cmpt 4461   Tr wtr 4496    |` cres 4953   ` cfv 5529   omcom 6589   reccrdg 6978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979
This theorem is referenced by:  epfrs  8066
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