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(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 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)
4	1, 2, 3	trcl 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((𝐴 ⊆ 𝑦 ∧ 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
7	5, 6	iunex 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) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∪ 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) → (𝑥 ⊆ 𝑦 ↔ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ⊆ 𝑦))
11	10	imbi2d 329	. . . . 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 1998	. . . 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 1548	. . 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 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((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 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