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Theorem tz9.1 9767
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9766 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 ∈ V
Assertion
Ref Expression
tz9.1 𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem tz9.1
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 9681 . . 3 ω ∈ V
2 fvex 6920 . . 3 ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
31, 2iunex 7992 . 2 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
4 sseq2 4022 . . 3 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝐴𝑥𝐴 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)))
5 treq 5273 . . 3 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (Tr 𝑥 ↔ Tr 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)))
6 sseq1 4021 . . . . 5 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝑥𝑦 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
76imbi2d 340 . . . 4 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦) ↔ ((𝐴𝑦 ∧ Tr 𝑦) → 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦)))
87albidv 1918 . . 3 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦) ↔ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦)))
94, 5, 83anbi123d 1435 . 2 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → ((𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦)) ↔ (𝐴 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ Tr 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))))
10 tz9.1.1 . . 3 𝐴 ∈ V
11 eqid 2735 . . 3 (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)
12 eqid 2735 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)
1310, 11, 12trcl 9766 . 2 (𝐴 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ Tr 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
143, 9, 13ceqsexv2d 3533 1 𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cun 3961  wss 3963   cuni 4912   ciun 4996  cmpt 5231  Tr wtr 5265  cres 5691  cfv 6563  ωcom 7887  reccrdg 8448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  epfrs  9769
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