Theorem tz9.13 3507

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(s):   x,A

Proof of Theorem tz9.13

Step	Hyp	Ref	Expression
1		tz9.13.1	. . 3 |- A e. V
2		setind 3492	. . . 4 |- (A.z(z (_ {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 1502	. . . . . . . 8 |- (z (_ {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 1350	. . . . . . . . 9 |- w e. V
5		eleq1 1149	. . . . . . . . . 10 |- (y = w -> (y e. (R1` x) <-> w e. (R1` x)))
6	5	birexdv 1220	. . . . . . . . 9 |- (y = w -> (E.x e. On y e. (R1` x) <-> E.x e. On w e. (R1` x)))
7	4, 6	elab 1415	. . . . . . . 8 |- (w e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On w e. (R1` x))
8	3, 7	syl6ib 185	. . . . . . 7 |- (z (_ {y | E.x e. On y e. (R1` x)} -> (w e. z -> E.x e. On w e. (R1` x)))
9	8	r19.21aiv 1259	. . . . . 6 |- (z (_ {y | E.x e. On y e. (R1` x)} -> A.w e. z E.x e. On w e. (R1` x))
10		visset 1350	. . . . . . 7 |- z e. V
11	10	tz9.12 3506	. . . . . 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 (_ {y | E.x e. On y e. (R1` x)} -> E.x e. On z e. (R1` x))
13		eleq1 1149	. . . . . . 7 |- (y = z -> (y e. (R1` x) <-> z e. (R1` x)))
14	13	birexdv 1220	. . . . . 6 |- (y = z -> (E.x e. On y e. (R1` x) <-> E.x e. On z e. (R1` x)))
15	10, 14	elab 1415	. . . . 5 |- (z e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On z e. (R1` x))
16	12, 15	sylibr 175	. . . 4 |- (z (_ {y | E.x e. On y e. (R1` x)} -> z e. {y | E.x e. On y e. (R1` x)})
17	2, 16	mpg 684	. . 3 |- {y | E.x e. On y e. (R1` x)} = V
18	1, 17	eleqtrr 1162	. 2 |- A e. {y | E.x e. On y e. (R1` x)}
19		eleq1 1149	. . . 4 |- (y = A -> (y e. (R1` x) <-> A e. (R1` x)))
20	19	birexdv 1220	. . 3 |- (y = A -> (E.x e. On y e. (R1` x) <-> E.x e. On A e. (R1` x)))
21	1, 20	elab 1415	. 2 |- (A e. {y | E.x e. On y e. (R1` x)} <-> E.x e. On A e. (R1` x))
22	18, 21	mpbi 164	1 |- E.x e. On A e. (R1` x)

Syntax hints:   -> wi 2   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  Oncon0 2199  ` cfv 2422  R1cr1 3485

This theorem is referenced by:  tz9.13g 3508  jech9.3 3510

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487

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