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Theorem tz9.1 7654
 Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7653 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 2435 . . 3
3 eqid 2435 . . 3
41, 2, 3trcl 7653 . 2
5 omex 7587 . . . 4
6 fvex 5733 . . . 4
75, 6iunex 5982 . . 3
8 sseq2 3362 . . . 4
9 treq 4300 . . . 4
10 sseq1 3361 . . . . . 6
1110imbi2d 308 . . . . 5
1211albidv 1635 . . . 4
138, 9, 123anbi123d 1254 . . 3
147, 13spcev 3035 . 2
154, 14ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936  wal 1549  wex 1550   wceq 1652   wcel 1725  cvv 2948   cun 3310   wss 3312  cuni 4007  ciun 4085   cmpt 4258   wtr 4294  com 4836   cres 4871  cfv 5445  crdg 6658 This theorem is referenced by:  epfrs  7656 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659
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