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Theorem tz9.1 9769
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9768 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 9683 . . 3 ω ∈ V
2 fvex 6919 . . 3 ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
31, 2iunex 7993 . 2 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
4 sseq2 4010 . . 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 5267 . . 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 4009 . . . . 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 1920 . . 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 1438 . 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 2737 . . 3 (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)
12 eqid 2737 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)
1310, 11, 12trcl 9768 . 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 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cun 3949  wss 3951   cuni 4907   ciun 4991  cmpt 5225  Tr wtr 5259  cres 5687  cfv 6561  ωcom 7887  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  epfrs  9771
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