Theorem tz9.1 8605

Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8604 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.

Step	Hyp	Ref	Expression
1		tz9.1.1	. . 3 |- A e. _V
2		eqid 2622	. . 3 |- ( rec ( ( w e. _V |-> ( w u. U. w ) ) , A ) |` om ) = ( rec ( ( w e. _V |-> ( w u. U. w ) ) , A ) |` om )
3		eqid 2622	. . 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 )
4	1, 2, 3	trcl 8604	. 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 8540	. . . 4 |- om e. _V
6		fvex 6201	. . . 4 |- ( ( rec ( ( w e. _V |-> ( w u. U. w ) ) , A ) |` om ) ` z ) e. _V
7	5, 6	iunex 7147	. . 3 |- U_ z e. om ( ( rec ( ( w e. _V |-> ( w u. U. w ) ) , A ) |` om ) ` z ) e. _V
8		sseq2 3627	. . . 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 4758	. . . 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 3626	. . . . . 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 ) )
11	10	imbi2d 330	. . . . 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 ) ) )
12	11	albidv 1849	. . . 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 ) ) )
13	8, 9, 12	3anbi123d 1399	. . 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 ) ) ) )
14	7, 13	spcev 3300	. 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 ) ) )
15	4, 14	ax-mp 5	1 |- E. x ( A C_ x /\ Tr x /\ A. y ( ( A C_ y /\ Tr y ) -> x C_ y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   Tr wtr 4752   |` cres 5116   ` cfv 5888   omcom 7065   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  epfrs  8607