MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz9.1 Structured version   Visualization version   GIF version

Theorem tz9.1 9670
Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9669 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 9584 . . 3 ω ∈ V
2 fvex 6856 . . 3 ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
31, 2iunex 7902 . 2 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
4 sseq2 3971 . . 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 5231 . . 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 3970 . . . . 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 341 . . . 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 1924 . . 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 1437 . 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 2733 . . 3 (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)
12 eqid 2733 . . 3 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧) = 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 𝑤)), 𝐴) ↾ ω)‘𝑧)
1310, 11, 12trcl 9669 . 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 3496 1 𝑥(𝐴𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴𝑦 ∧ Tr 𝑦) → 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  cun 3909  wss 3911   cuni 4866   ciun 4955  cmpt 5189  Tr wtr 5223  cres 5636  cfv 6497  ωcom 7803  reccrdg 8356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357
This theorem is referenced by:  epfrs  9672
  Copyright terms: Public domain W3C validator