Theorem rankval4 4712

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
rankr1b.1	⊢ A ∈ V

Assertion

Ref	Expression
rankval4	⊢ (rank ‘A) = ∪x ∈ A suc (rank ‘x)

Distinct variable group:   x,A

Proof of Theorem rankval4

Step	Hyp	Ref	Expression
1		ax-17 973	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 973	. . . . . . 7 ⊢ (y ∈ R1 → ∀x y ∈ R1)
3		hbiu1 2588	. . . . . . 7 ⊢ (y ∈ ∪x ∈ A suc (rank ‘x) → ∀x y ∈ ∪x ∈ A suc (rank ‘x))
4	2, 3	hbfv 3735	. . . . . 6 ⊢ (y ∈ (R1 ‘∪x ∈ A suc (rank ‘x)) → ∀x y ∈ (R1 ‘∪x ∈ A suc (rank ‘x)))
5	1, 4	dfss2f 2063	. . . . 5 ⊢ (A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)) ↔ ∀x(x ∈ A → x ∈ (R1 ‘∪x ∈ A suc (rank ‘x))))
6		visset 1816	. . . . . . 7 ⊢ x ∈ V
7	6	rankid 4682	. . . . . 6 ⊢ x ∈ (R1 ‘suc (rank ‘x))
8		ssiun2 2597	. . . . . . . 8 ⊢ (x ∈ A → suc (rank ‘x) ⊆ ∪x ∈ A suc (rank ‘x))
9		rankon 4681	. . . . . . . . . 10 ⊢ (rank ‘x) ∈ On
10	9	onsuc 3111	. . . . . . . . 9 ⊢ suc (rank ‘x) ∈ On
11		rankr1b.1	. . . . . . . . . . 11 ⊢ A ∈ V
12		fvex 3738	. . . . . . . . . . . 12 ⊢ (rank ‘x) ∈ V
13	12	sucex 3056	. . . . . . . . . . 11 ⊢ suc (rank ‘x) ∈ V
14	11, 13	iunon 3915	. . . . . . . . . 10 ⊢ (∀x ∈ A suc (rank ‘x) ∈ On → ∪x ∈ A suc (rank ‘x) ∈ On)
15	10	a1i 8	. . . . . . . . . 10 ⊢ (x ∈ A → suc (rank ‘x) ∈ On)
16	14, 15	mprg 1703	. . . . . . . . 9 ⊢ ∪x ∈ A suc (rank ‘x) ∈ On
17		r1ord3 4667	. . . . . . . . 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 699	. . . . . . . 8 ⊢ (suc (rank ‘x) ⊆ ∪x ∈ A suc (rank ‘x) → (R1 ‘suc (rank ‘x)) ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)))
19	8, 18	syl 10	. . . . . . 7 ⊢ (x ∈ A → (R1 ‘suc (rank ‘x)) ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)))
20	19	sseld 2070	. . . . . 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 990	. . . 4 ⊢ A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x))
23		fvex 3738	. . . . 5 ⊢ (R1 ‘∪x ∈ A suc (rank ‘x)) ∈ V
24	23	rankss 4698	. . . 4 ⊢ (A ⊆ (R1 ‘∪x ∈ A suc (rank ‘x)) → (rank ‘A) ⊆ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))))
25	22, 24	ax-mp 7	. . 3 ⊢ (rank ‘A) ⊆ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x)))
26		r1ord3 4667	. . . . . . 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 697	. . . . . 6 ⊢ (y ∈ On → (∪x ∈ A suc (rank ‘x) ⊆ y → (R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)))
28	27	ss2rabi 2131	. . . . 5 ⊢ {y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y} ⊆ {y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)}
29		intss 2558	. . . . 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 7	. . . 4 ⊢ ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)} ⊆ ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y}
31		rankval2 4680	. . . . 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 7	. . . 4 ⊢ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))) = ∩{y ∈ On∣(R1 ‘∪x ∈ A suc (rank ‘x)) ⊆ (R1 ‘y)}
33		intmin 2557	. . . . . 6 ⊢ (∪x ∈ A suc (rank ‘x) ∈ On → ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y} = ∪x ∈ A suc (rank ‘x))
34	16, 33	ax-mp 7	. . . . 5 ⊢ ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y} = ∪x ∈ A suc (rank ‘x)
35	34	eqcomi 1482	. . . 4 ⊢ ∪x ∈ A suc (rank ‘x) = ∩{y ∈ On∣∪x ∈ A suc (rank ‘x) ⊆ y}
36	30, 32, 35	3sstr4 2103	. . 3 ⊢ (rank ‘(R1 ‘∪x ∈ A suc (rank ‘x))) ⊆ ∪x ∈ A suc (rank ‘x)
37	25, 36	sstri 2076	. 2 ⊢ (rank ‘A) ⊆ ∪x ∈ A suc (rank ‘x)
38		iunss 2595	. . 3 ⊢ (∪x ∈ A suc (rank ‘x) ⊆ (rank ‘A) ↔ ∀x ∈ A suc (rank ‘x) ⊆ (rank ‘A))
39	11	rankel 4690	. . . 4 ⊢ (x ∈ A → (rank ‘x) ∈ (rank ‘A))
40		rankon 4681	. . . . 5 ⊢ (rank ‘A) ∈ On
41	9, 40	onsucss 3117	. . . 4 ⊢ ((rank ‘x) ∈ (rank ‘A) ↔ suc (rank ‘x) ⊆ (rank ‘A))
42	39, 41	sylib 198	. . 3 ⊢ (x ∈ A → suc (rank ‘x) ⊆ (rank ‘A))
43	38, 42	mprgbir 1704	. 2 ⊢ ∪x ∈ A suc (rank ‘x) ⊆ (rank ‘A)
44	37, 43	eqssi 2081	1 ⊢ (rank ‘A) = ∪x ∈ A suc (rank ‘x)

Syntax hints:   → wi 3   = wceq 958   ∈ wcel 960  {crab 1651  Vcvv 1814   ⊆ wss 2050  ∩cint 2537  ∪ciun 2570  Oncon0 2954  suc csuc 2956   ‘cfv 3188  R₁cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankbnd 4713  rankc1 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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