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Theorem tz9.1 8181
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8180 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
Assertion
Ref Expression
tz9.1
Distinct variable group:   , ,

Proof of Theorem tz9.1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.1.1 . . 3
2 eqid 2457 . . 3
3 eqid 2457 . . 3
41, 2, 3trcl 8180 . 2
5 omex 8081 . . . 4
6 fvex 5881 . . . 4
75, 6iunex 6780 . . 3
8 sseq2 3525 . . . 4
9 treq 4551 . . . 4
10 sseq1 3524 . . . . . 6
1110imbi2d 316 . . . . 5
1211albidv 1713 . . . 4
138, 9, 123anbi123d 1299 . . 3
147, 13spcev 3201 . 2
154, 14ax-mp 5 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  u.cun 3473  C_wss 3475  U.cuni 4249  U_ciun 4330  e.cmpt 4510  Trwtr 4545  |`cres 5006  `cfv 5593   com 6700  reccrdg 7094
This theorem is referenced by:  epfrs  8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095
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