Theorem tz9.1 8855

Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 8854 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		tz9.1.1	. . 3 ⊢ 𝐴 ∈ V
2		eqid 2799	. . 3 ⊢ (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)
3		eqid 2799	. . 3 ⊢ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧)
4	1, 2, 3	trcl 8854	. 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))
5		omex 8790	. . . 4 ⊢ ω ∈ V
6		fvex 6424	. . . 4 ⊢ ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
7	5, 6	iunex 7381	. . 3 ⊢ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
8		sseq2 3823	. . . 4 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧)))
9		treq 4951	. . . 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 3822	. . . . . 6 ⊢ (𝑥 = ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) → (𝑥 ⊆ 𝑦 ↔ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
11	10	imbi2d 332	. . . . 5 ⊢ (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)))
12	11	albidv 2016	. . . 4 ⊢ (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)))
13	8, 9, 12	3anbi123d 1561	. . 3 ⊢ (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))))
14	7, 13	spcev 3488	. 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)) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
15	4, 14	ax-mp 5	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108  ∀wal 1651   = wceq 1653  ∃wex 1875   ∈ wcel 2157  Vcvv 3385   ∪ cun 3767   ⊆ wss 3769  ∪ cuni 4628  ∪ ciun 4710   ↦ cmpt 4922  Tr wtr 4945   ↾ cres 5314  ‘cfv 6101  ωcom 7299  reccrdg 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745

This theorem is referenced by:  epfrs  8857