Theorem ranklon 3540

Description: The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204.

Hypothesis

Ref	Expression
ranklon.1	⊢ A ∈ V

Assertion

Ref	Expression
ranklon	⊢ (rank ‘A) = ∪x ∈ A suc (rank ‘x)

Distinct variable group(s):   x,A

Proof of Theorem ranklon

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. . . . . . 7 ⊢ (y ∈ R1 → ∀x y ∈ R1)
3		hbiu1 2012	. . . . . . 7 ⊢ (y ∈ ∪x ∈ A suc (rank ‘x) → ∀x y ∈ ∪x ∈ A suc (rank ‘x))
4	2, 3	hbfv 2837	. . . . . 6 ⊢ (y ∈ (R1 ‘∪x ∈ A suc (rank ‘x)) → ∀x y ∈ (R1 ‘∪x ∈ A suc (rank ‘x)))
5	1, 4	dfss2f 1499	. . . . 5 ⊢ (A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)) ↔ ∀x(x ∈ A → x ∈ (R1 ‘∪x ∈ A suc (rank ‘x))))
6		visset 1350	. . . . . . 7 ⊢ x ∈ V
7	6	rankid 3516	. . . . . 6 ⊢ x ∈ (R1 ‘suc (rank ‘x))
8		ssiun2 2019	. . . . . . . 8 ⊢ (x ∈ A → suc (rank ‘x) ⊆ ∪x ∈ A suc (rank ‘x))
9		rankon 3515	. . . . . . . . . 10 ⊢ (rank ‘x) ∈ On
10	9	onsuc 2353	. . . . . . . . 9 ⊢ suc (rank ‘x) ∈ On
11		ranklon.1	. . . . . . . . . . 11 ⊢ A ∈ V
12		fvex 2838	. . . . . . . . . . . 12 ⊢ (rank ‘x) ∈ V
13	12	sucex 2303	. . . . . . . . . . 11 ⊢ suc (rank ‘x) ∈ V
14	11, 13	iunon 2947	. . . . . . . . . 10 ⊢ (∀x ∈ A suc (rank ‘x) ∈ On → ∪x ∈ A suc (rank ‘x) ∈ On)
15	10	a1i 7	. . . . . . . . . 10 ⊢ (x ∈ A → suc (rank ‘x) ∈ On)
16	14, 15	mprg 1249	. . . . . . . . 9 ⊢ ∪x ∈ A suc (rank ‘x) ∈ On
17		r1ord3 3501	. . . . . . . . 9 ⊢ ((suc (rank ‘x) ∈ On ∧ ∪x ∈ A suc (rank ‘x) ∈ On) → (suc (rank ‘x) ⊆ ∪x ∈ A suc (rank ‘x) → (R1 ‘suc (rank ‘x)) ⊆ (R1 ‘∪x ∈ A suc (rank ‘x))))
18	10, 16, 17	mp2an 520	. . . . . . . 8 ⊢ (suc (rank ‘x) ⊆ ∪x ∈ A suc (rank ‘x) → (R1 ‘suc (rank ‘x)) ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)))
19	8, 18	syl 12	. . . . . . 7 ⊢ (x ∈ A → (R1 ‘suc (rank ‘x)) ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)))
20	19	sseld 1506	. . . . . 6 ⊢ (x ∈ A → (x ∈ (R1 ‘suc (rank ‘x)) → x ∈ (R1 ‘∪x ∈ A suc (rank ‘x))))
21	7, 20	mpi 44	. . . . 5 ⊢ (x ∈ A → x ∈ (R1 ‘∪x ∈ A suc (rank ‘x)))
22	5, 21	mpgbir 686	. . . 4 ⊢ A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x))
23		fvex 2838	. . . . 5 ⊢ (R1 ‘∪x ∈ A suc (rank ‘x)) ∈ V
24	23	rankss 3531	. . . 4 ⊢ (A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)) → (rank ‘A) ⊆ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))))
25	22, 24	ax-mp 6	. . 3 ⊢ (rank ‘A) ⊆ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x)))
26		r1ord3 3501	. . . . . . 7 ⊢ ((∪x ∈ A suc (rank ‘x) ∈ On ∧ y ∈ On) → (∪x ∈ A suc (rank ‘x) ⊆ y → (R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)))
27	16, 26	mpan 518	. . . . . 6 ⊢ (y ∈ On → (∪x ∈ A suc (rank ‘x) ⊆ y → (R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)))
28	27	ss2rabi 1554	. . . . 5 ⊢ {y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y} ⊆ {y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)}
29		intss 1983	. . . . 5 ⊢ ({y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y} ⊆ {y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)} → ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)} ⊆ ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y})
30	28, 29	ax-mp 6	. . . 4 ⊢ ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)} ⊆ ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y}
31		rankval2 3514	. . . . 5 ⊢ ((R1 ‘∪x ∈ A suc (rank ‘x)) ∈ V → (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))) = ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)})
32	23, 31	ax-mp 6	. . . 4 ⊢ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))) = ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)}
33		intmin 1982	. . . . 5 ⊢ (∪x ∈ A suc (rank ‘x) ∈ On → ∪x ∈ A suc (rank ‘x) = ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y})
34	16, 33	ax-mp 6	. . . 4 ⊢ ∪x ∈ A suc (rank ‘x) = ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y}
35	30, 32, 34	3sstr4 1539	. . 3 ⊢ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))) ⊆ ∪x ∈ A suc (rank ‘x)
36	25, 35	sstri 1512	. 2 ⊢ (rank ‘A) ⊆ ∪x ∈ A suc (rank ‘x)
37		iunss 2017	. . 3 ⊢ (∪x ∈ A suc (rank ‘x) ⊆ (rank ‘A) ↔ ∀x ∈ A suc (rank ‘x) ⊆ (rank ‘A))
38	11	rankel 3524	. . . 4 ⊢ (x ∈ A → (rank ‘x) ∈ (rank ‘A))
39		rankon 3515	. . . . 5 ⊢ (rank ‘A) ∈ On
40	9, 39	onsucss 2359	. . . 4 ⊢ ((rank ‘x) ∈ (rank ‘A) ↔ suc (rank ‘x) ⊆ (rank ‘A))
41	38, 40	sylib 173	. . 3 ⊢ (x ∈ A → suc (rank ‘x) ⊆ (rank ‘A))
42	37, 41	mprgbir 1250	. 2 ⊢ ∪x ∈ A suc (rank ‘x) ⊆ (rank ‘A)
43	36, 42	eqssi 1517	1 ⊢ (rank ‘A) = ∪x ∈ A suc (rank ‘x)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∩cint 1965  ∪ciun 1994  Oncon0 2199  suc csuc 2201   ‘cfv 2422  R₁cr1 3485  rankcrnk 3486

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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