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 ∈ V

Assertion

Ref	Expression
tz9.13	⊢ ∃x ∈ On A ∈ (R1 ‘x)

Distinct variable group(s):   x,A

Proof of Theorem tz9.13

Step	Hyp	Ref	Expression
1		tz9.13.1	. . 3 ⊢ A ∈ V
2		setind 3492	. . . 4 ⊢ (∀z(z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → z ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)}) → {y∣∃x ∈ On y ∈ (R1 ‘x)} = V)
3		ssel 1502	. . . . . . . 8 ⊢ (z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → (w ∈ z → w ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)}))
4		visset 1350	. . . . . . . . 9 ⊢ w ∈ V
5		eleq1 1149	. . . . . . . . . 10 ⊢ (y = w → (y ∈ (R1 ‘x) ↔ w ∈ (R1 ‘x)))
6	5	birexdv 1220	. . . . . . . . 9 ⊢ (y = w → (∃x ∈ On y ∈ (R1 ‘x) ↔ ∃x ∈ On w ∈ (R1 ‘x)))
7	4, 6	elab 1415	. . . . . . . 8 ⊢ (w ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)} ↔ ∃x ∈ On w ∈ (R1 ‘x))
8	3, 7	syl6ib 185	. . . . . . 7 ⊢ (z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → (w ∈ z → ∃x ∈ On w ∈ (R1 ‘x)))
9	8	r19.21aiv 1259	. . . . . 6 ⊢ (z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → ∀w ∈ z ∃x ∈ On w ∈ (R1 ‘x))
10		visset 1350	. . . . . . 7 ⊢ z ∈ V
11	10	tz9.12 3506	. . . . . 6 ⊢ (∀w ∈ z ∃x ∈ On w ∈ (R1 ‘x) → ∃x ∈ On z ∈ (R1 ‘x))
12	9, 11	syl 12	. . . . 5 ⊢ (z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → ∃x ∈ On z ∈ (R1 ‘x))
13		eleq1 1149	. . . . . . 7 ⊢ (y = z → (y ∈ (R1 ‘x) ↔ z ∈ (R1 ‘x)))
14	13	birexdv 1220	. . . . . 6 ⊢ (y = z → (∃x ∈ On y ∈ (R1 ‘x) ↔ ∃x ∈ On z ∈ (R1 ‘x)))
15	10, 14	elab 1415	. . . . 5 ⊢ (z ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)} ↔ ∃x ∈ On z ∈ (R1 ‘x))
16	12, 15	sylibr 175	. . . 4 ⊢ (z ⊆ {y∣∃x ∈ On y ∈ (R1 ‘x)} → z ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)})
17	2, 16	mpg 684	. . 3 ⊢ {y∣∃x ∈ On y ∈ (R1 ‘x)} = V
18	1, 17	eleqtrr 1162	. 2 ⊢ A ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)}
19		eleq1 1149	. . . 4 ⊢ (y = A → (y ∈ (R1 ‘x) ↔ A ∈ (R1 ‘x)))
20	19	birexdv 1220	. . 3 ⊢ (y = A → (∃x ∈ On y ∈ (R1 ‘x) ↔ ∃x ∈ On A ∈ (R1 ‘x)))
21	1, 20	elab 1415	. 2 ⊢ (A ∈ {y∣∃x ∈ On y ∈ (R1 ‘x)} ↔ ∃x ∈ On A ∈ (R1 ‘x))
22	18, 21	mpbi 164	1 ⊢ ∃x ∈ On A ∈ (R1 ‘x)

Syntax hints:   → wi 2   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃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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