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Theorem tz9.1 7295
 Description: Every set has a transitive closure (smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 7294 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
StepHypRef Expression
1 tz9.1.1 . . 3
2 eqid 2253 . . 3
3 eqid 2253 . . 3
41, 2, 3trcl 7294 . 2
5 omex 7228 . . . 4
6 fvex 5391 . . . 4
75, 6iunex 5622 . . 3
8 sseq2 3121 . . . 4
9 treq 4016 . . . 4
10 sseq1 3120 . . . . . 6
1110imbi2d 309 . . . . 5
1211albidv 2004 . . . 4
138, 9, 123anbi123d 1257 . . 3
147, 13cla4ev 2812 . 2
154, 14ax-mp 10 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   w3a 939  wal 1532  wex 1537   wceq 1619   wcel 1621  cvv 2727   cun 3076   wss 3078  cuni 3727  ciun 3803   cmpt 3974   wtr 4010  com 4547   cres 4582  cfv 4592  crdg 6308 This theorem is referenced by:  epfrs  7297 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-inf2 7226 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309
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