Theorem rankval3 3525

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79.

Hypothesis

Ref	Expression
rankval3.1	⊢ A ∈ V

Assertion

Ref	Expression
rankval3	⊢ (rank ‘A) = ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x}

Distinct variable group(s):   x,y,A

Proof of Theorem rankval3

Step	Hyp	Ref	Expression
1		rankval3.1	. . . 4 ⊢ A ∈ V
2	1	rankval 3512	. . 3 ⊢ (rank ‘A) = ∩{x ∈ On∣A ∈ (R1 ‘suc x)}
3		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
4	3	rankid 3516	. . . . . . . 8 ⊢ y ∈ (R1 ‘suc (rank ‘y))
5		eloni 2209	. . . . . . . . . . 11 ⊢ (x ∈ On → Ord x)
6		ordsucss 2320	. . . . . . . . . . 11 ⊢ (Ord x → ((rank ‘y) ∈ x → suc (rank ‘y) ⊆ x))
7	5, 6	syl 12	. . . . . . . . . 10 ⊢ (x ∈ On → ((rank ‘y) ∈ x → suc (rank ‘y) ⊆ x))
8		rankon 3515	. . . . . . . . . . . 12 ⊢ (rank ‘y) ∈ On
9	8	onsuc 2353	. . . . . . . . . . 11 ⊢ suc (rank ‘y) ∈ On
10		r1ord3 3501	. . . . . . . . . . 11 ⊢ ((suc (rank ‘y) ∈ On ∧ x ∈ On) → (suc (rank ‘y) ⊆ x → (R1 ‘suc (rank ‘y)) ⊆ (R1 ‘x)))
11	9, 10	mpan 518	. . . . . . . . . 10 ⊢ (x ∈ On → (suc (rank ‘y) ⊆ x → (R1 ‘suc (rank ‘y)) ⊆ (R1 ‘x)))
12	7, 11	syld 27	. . . . . . . . 9 ⊢ (x ∈ On → ((rank ‘y) ∈ x → (R1 ‘suc (rank ‘y)) ⊆ (R1 ‘x)))
13		ssel 1502	. . . . . . . . 9 ⊢ ((R1 ‘suc (rank ‘y)) ⊆ (R1 ‘x) → (y ∈ (R1 ‘suc (rank ‘y)) → y ∈ (R1 ‘x)))
14	12, 13	syl6 23	. . . . . . . 8 ⊢ (x ∈ On → ((rank ‘y) ∈ x → (y ∈ (R1 ‘suc (rank ‘y)) → y ∈ (R1 ‘x))))
15	4, 14	mpii 45	. . . . . . 7 ⊢ (x ∈ On → ((rank ‘y) ∈ x → y ∈ (R1 ‘x)))
16	15	r19.20sdv 1257	. . . . . 6 ⊢ (x ∈ On → (∀y ∈ A (rank ‘y) ∈ x → ∀y ∈ A y ∈ (R1 ‘x)))
17		r1suc 3496	. . . . . . . 8 ⊢ (x ∈ On → (R1 ‘suc x) = ℘(R1 ‘x))
18	17	eleq2d 1156	. . . . . . 7 ⊢ (x ∈ On → (A ∈ (R1 ‘suc x) ↔ A ∈ ℘(R1 ‘x)))
19	1	elpw 1801	. . . . . . . 8 ⊢ (A ∈ ℘(R1 ‘x) ↔ A ⊆ (R1 ‘x))
20		dfss3 1498	. . . . . . . 8 ⊢ (A ⊆ (R1 ‘x) ↔ ∀y ∈ A y ∈ (R1 ‘x))
21	19, 20	bitr 151	. . . . . . 7 ⊢ (A ∈ ℘(R1 ‘x) ↔ ∀y ∈ A y ∈ (R1 ‘x))
22	18, 21	syl6bb 414	. . . . . 6 ⊢ (x ∈ On → (A ∈ (R1 ‘suc x) ↔ ∀y ∈ A y ∈ (R1 ‘x)))
23	16, 22	sylibrd 179	. . . . 5 ⊢ (x ∈ On → (∀y ∈ A (rank ‘y) ∈ x → A ∈ (R1 ‘suc x)))
24	23	ss2rabi 1554	. . . 4 ⊢ {x ∈ On∣∀y ∈ A (rank ‘y) ∈ x} ⊆ {x ∈ On∣A ∈ (R1 ‘suc x)}
25		intss 1983	. . . 4 ⊢ ({x ∈ On∣∀y ∈ A (rank ‘y) ∈ x} ⊆ {x ∈ On∣A ∈ (R1 ‘suc x)} → ∩{x ∈ On∣A ∈ (R1 ‘suc x)} ⊆ ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x})
26	24, 25	ax-mp 6	. . 3 ⊢ ∩{x ∈ On∣A ∈ (R1 ‘suc x)} ⊆ ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x}
27	2, 26	eqsstr 1530	. 2 ⊢ (rank ‘A) ⊆ ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x}
28		rankon 3515	. . 3 ⊢ (rank ‘A) ∈ On
29	1	rankel 3524	. . . 4 ⊢ (y ∈ A → (rank ‘y) ∈ (rank ‘A))
30	29	rgen 1247	. . 3 ⊢ ∀y ∈ A (rank ‘y) ∈ (rank ‘A)
31		eleq2 1150	. . . . 5 ⊢ (x = (rank ‘A) → ((rank ‘y) ∈ x ↔ (rank ‘y) ∈ (rank ‘A)))
32	31	biraldv 1219	. . . 4 ⊢ (x = (rank ‘A) → (∀y ∈ A (rank ‘y) ∈ x ↔ ∀y ∈ A (rank ‘y) ∈ (rank ‘A)))
33	32	onintss 2266	. . 3 ⊢ ((rank ‘A) ∈ On → (∀y ∈ A (rank ‘y) ∈ (rank ‘A) → ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x} ⊆ (rank ‘A)))
34	28, 30, 33	mp2 43	. 2 ⊢ ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x} ⊆ (rank ‘A)
35	27, 34	eqssi 1517	1 ⊢ (rank ‘A) = ∩{x ∈ On∣∀y ∈ A (rank ‘y) ∈ x}

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  ∩cint 1965  Ord word 2198  Oncon0 2199  suc csuc 2201   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

This theorem is referenced by:  ranksn 3532  rankuni 3533  rankun 3535  rankonid 3538

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  df-rank 3488

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