Theorem tz9.1 9165

Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9164 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(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem tz9.1

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 9100	. . 3 ⊢ ω ∈ V
2		fvex 6678	. . 3 ⊢ ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
3	1, 2	iunex 7663	. 2 ⊢ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
4		sseq2 3993	. . 3 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧)))
5		treq 5171	. . 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 3992	. . . . 5 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝑥 ⊆ 𝑦 ↔ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
7	6	imbi2d 343	. . . 4 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦)))
8	7	albidv 1917	. . 3 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦)))
9	4, 5, 8	3anbi123d 1432	. 2 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) ↔ (𝐴 ⊆ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ Tr ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))))
10		tz9.1.1	. . 3 ⊢ 𝐴 ∈ V
11		eqid 2821	. . 3 ⊢ (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)
12		eqid 2821	. . 3 ⊢ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧)
13	10, 11, 12	trcl 9164	. 2 ⊢ (𝐴 ⊆ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ Tr ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
14	3, 9, 13	ceqsexv2d 3543	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3495   ∪ cun 3934   ⊆ wss 3936  ∪ cuni 4832  ∪ ciun 4912   ↦ cmpt 5139  Tr wtr 5165   ↾ cres 5552  ‘cfv 6350  ωcom 7574  reccrdg 8039

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040

This theorem is referenced by:  epfrs  9167