Theorem tz9.13 5774

Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78.

Hypothesis

Ref	Expression
tz9.13.1	|- A e. _V

Assertion

Ref	Expression
tz9.13	|- E.x e. On A e. (R1` x)

Distinct variable group:   x,A

Proof of Theorem tz9.13

Step	Hyp	Ref	Expression
1		tz9.13.1	. . 3 |- A e. _V
2		setind 5759	. . . 4 |- (A.z(z C_ {y | E.x e. On y e. (R1` x)} -> z e. {y | E.x e. On y e. (R1` x)}) -> {y | E.x e. On y e. (R1` x)} = _V)
3		ssel 2615	. . . . . . . 8 |- (z C_ {y | E.x e. On y e. (R1` x)} -> (w e. z -> w e. {y | E.x e. On y e. (R1` x)}))
4		visset 2295	. . . . . . . . 9 |- w e. _V
5		eleq1 1957	. . . . . . . . . 10 |- (y = w -> (y e. (R1` x) <-> w e. (R1` x)))
6	5	rexbidv 2124	. . . . . . . . 9 |- (y = w -> (E.x e. On y e. (R1` x) <-> E.x e. On w e. (R1` x)))
7	4, 6	elab 2403	. . . . . . . 8 |- (w e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On w e. (R1` x))
8	3, 7	syl6ib 229	. . . . . . 7 |- (z C_ {y | E.x e. On y e. (R1` x)} -> (w e. z -> E.x e. On w e. (R1` x)))
9	8	r19.21aiv 2175	. . . . . 6 |- (z C_ {y | E.x e. On y e. (R1` x)} -> A.w e. z E.x e. On w e. (R1` x))
10		visset 2295	. . . . . . 7 |- z e. _V
11	10	tz9.12 5773	. . . . . 6 |- (A.w e. z E.x e. On w e. (R1` x) -> E.x e. On z e. (R1` x))
12	9, 11	syl 12	. . . . 5 |- (z C_ {y | E.x e. On y e. (R1` x)} -> E.x e. On z e. (R1` x))
13		eleq1 1957	. . . . . . 7 |- (y = z -> (y e. (R1` x) <-> z e. (R1` x)))
14	13	rexbidv 2124	. . . . . 6 |- (y = z -> (E.x e. On y e. (R1` x) <-> E.x e. On z e. (R1` x)))
15	10, 14	elab 2403	. . . . 5 |- (z e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On z e. (R1` x))
16	12, 15	sylibr 217	. . . 4 |- (z C_ {y | E.x e. On y e. (R1` x)} -> z e. {y | E.x e. On y e. (R1` x)})
17	2, 16	mpg 1332	. . 3 |- {y | E.x e. On y e. (R1` x)} = _V
18	1, 17	eleqtrri 1970	. 2 |- A e. {y | E.x e. On y e. (R1` x)}
19		eleq1 1957	. . . 4 |- (y = A -> (y e. (R1` x) <-> A e. (R1` x)))
20	19	rexbidv 2124	. . 3 |- (y = A -> (E.x e. On y e. (R1` x) <-> E.x e. On A e. (R1` x)))
21	1, 20	elab 2403	. 2 |- (A e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On A e. (R1` x))
22	18, 21	mpbi 206	1 |- E.x e. On A e. (R1` x)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  Oncon0 3657  ` cfv 3998  R1cr1 5748

This theorem is referenced by:  tz9.13g 5775  jech9.3 5777

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-reg 5695  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-int 3215  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  df-r1 5750

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