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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
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 2443 . . 3
3 eqid 2443 . . 3
41, 2, 3trcl 8162 . 2
5 omex 8063 . . . 4
6 fvex 5866 . . . 4
75, 6iunex 6765 . . 3
8 sseq2 3511 . . . 4
9 treq 4536 . . . 4
10 sseq1 3510 . . . . . 6
1110imbi2d 316 . . . . 5
1211albidv 1700 . . . 4
138, 9, 123anbi123d 1300 . . 3
147, 13spcev 3187 . 2
154, 14ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 974  wal 1381   wceq 1383  wex 1599   wcel 1804  cvv 3095   cun 3459   wss 3461  cuni 4234  ciun 4315   cmpt 4495   wtr 4530   cres 4991  cfv 5578  com 6685  crdg 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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