Theorem tz9.1 9684

Description: Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9683 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 9598	. . 3 ⊢ ω ∈ V
2		fvex 6880	. . 3 ⊢ ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
3	1, 2	iunex 7949	. 2 ⊢ ∪ 𝑧 ∈ ω ((rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)‘𝑧) ∈ V
4		sseq2 3962	. . 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 5214	. . 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 3961	. . . . 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 342	. . . 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 1940	. . 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 1457	. 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 2762	. . 3 ⊢ (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω) = (rec((𝑤 ∈ V ↦ (𝑤 ∪ ∪ 𝑤)), 𝐴) ↾ ω)
12		eqid 2762	. . 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 9683	. 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 3503	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4865  ∪ ciun 4949   ↦ cmpt 5181  Tr wtr 5207   ↾ cres 5649  ‘cfv 6521  ωcom 7846  reccrdg 8380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381

This theorem is referenced by:  epfrs  9686  axtco  36831  ttcexg  36892