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Theorem tz9.1 8466
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8465 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 tz9.1.1 . . 3 𝐴 ∈ V
2 eqid 2610 . . 3 (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)
3 eqid 2610 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)
41, 2, 3trcl 8465 . 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))
5 omex 8401 . . . 4 ω ∈ V
6 fvex 6098 . . . 4 ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
75, 6iunex 7017 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
8 sseq2 3590 . . . 4 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝐴𝑥𝐴 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)))
9 treq 4681 . . . 4 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (Tr 𝑥 ↔ Tr 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)))
10 sseq1 3589 . . . . . 6 (𝑥 = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝑥𝑦 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
1110imbi2d 329 . . . . 5 (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)))
1211albidv 1836 . . . 4 (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)))
138, 9, 123anbi123d 1391 . . 3 (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))))
147, 13spcev 3273 . 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)) → ∃𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦)))
154, 14ax-mp 5 1 𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wal 1473   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  cun 3538  wss 3540   cuni 4367   ciun 4450  cmpt 4638  Tr wtr 4675  cres 5030  cfv 5790  ωcom 6935  reccrdg 7370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371
This theorem is referenced by:  epfrs  8468
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