Theorem tz9.1 5753

Description: Every set has a transitive closure (smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 5752 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.)

Hypothesis

Ref	Expression
tz9.1.1	|- A e. _V

Assertion

Ref	Expression
tz9.1	|- E.x(A C_ x /\ Tr x /\ A.y((A C_ y /\ Tr y) -> x C_ y))

Distinct variable group:   x,A,y

Proof of Theorem tz9.1

Step	Hyp	Ref	Expression
1		tz9.1.1	. . 3 |- A e. _V
2		eqid 1884	. . 3 |- (rec({<.w, v>. | v = (w u. U.w)}, A) |` om) = (rec({<.w, v>. | v = (w u. U.w)}, A) |` om)
3		eqid 1884	. . 3 |- U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z)
4	1, 2, 3	trcl 5752	. 2 |- (A C_ U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ Tr U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ A.y((A C_ y /\ Tr y) -> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y))
5		omex 5733	. . . 4 |- om e. _V
6		fvex 4689	. . . 4 |- ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) e. _V
7	5, 6	iunex 4839	. . 3 |- U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) e. _V
8		sseq2 2639	. . . 4 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> (A C_ x <-> A C_ U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z)))
9		treq 3417	. . . 4 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> (Tr x <-> Tr U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z)))
10		sseq1 2637	. . . . . 6 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> (x C_ y <-> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y))
11	10	imbi2d 674	. . . . 5 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> (((A C_ y /\ Tr y) -> x C_ y) <-> ((A C_ y /\ Tr y) -> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y)))
12	11	albidv 1656	. . . 4 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> (A.y((A C_ y /\ Tr y) -> x C_ y) <-> A.y((A C_ y /\ Tr y) -> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y)))
13	8, 9, 12	3anbi123d 1168	. . 3 |- (x = U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) -> ((A C_ x /\ Tr x /\ A.y((A C_ y /\ Tr y) -> x C_ y)) <-> (A C_ U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ Tr U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ A.y((A C_ y /\ Tr y) -> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y))))
14	7, 13	cla4ev 2371	. 2 |- ((A C_ U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ Tr U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) /\ A.y((A C_ y /\ Tr y) -> U_z e. om ((rec({<.w, v>. | v = (w u. U.w)}, A) |` om)` z) C_ y)) -> E.x(A C_ x /\ Tr x /\ A.y((A C_ y /\ Tr y) -> x C_ y)))
15	4, 14	ax-mp 7	1 |- E.x(A C_ x /\ Tr x /\ A.y((A C_ y /\ Tr y) -> x C_ y))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   u. cun 2591   C_ wss 2593  U.cuni 3177  U_ciun 3255  {copab 3395  Tr wtr 3411  omcom 3949   |` cres 3988  ` cfv 3998  reccrdg 5139

This theorem is referenced by:  zfregs 5754

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

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