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Theorem tz9.1 8778
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8777 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 2760 . . 3 (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)
3 eqid 2760 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)
41, 2, 3trcl 8777 . 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 8713 . . . 4 ω ∈ V
6 fvex 6362 . . . 4 ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
75, 6iunex 7312 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
8 sseq2 3768 . . . 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 4910 . . . 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 3767 . . . . . 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 1998 . . . 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 1548 . . 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 3440 . 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 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cun 3713  wss 3715   cuni 4588   ciun 4672  cmpt 4881  Tr wtr 4904  cres 5268  cfv 6049  ωcom 7230  reccrdg 7674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675
This theorem is referenced by:  epfrs  8780
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