Theorem rankval4 9625

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. (Contributed by NM, 12-Oct-2003.)

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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝐴
2		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
3		nfiu1 4958	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
4	2, 3	nffv 6784	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
5	1, 4	dfss2f 3911	. . . . 5 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
6		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankid 9591	. . . . . 6 ⊢ 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8		ssiun2 4977	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
9		rankon 9553	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
10	9	onsuci 7685	. . . . . . . . 9 ⊢ suc (rank‘𝑥) ∈ On
11		rankr1b.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
12	10	rgenw 3076	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
13		iunon 8170	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
14	11, 12, 13	mp2an 689	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
15		r1ord3 9540	. . . . . . . . 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))))
16	10, 14, 15	mp2an 689	. . . . . . . 8 ⊢ (suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
18	17	sseld 3920	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝑅1‘suc (rank‘𝑥)) → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
19	7, 18	mpi 20	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
20	5, 19	mpgbir 1802	. . . 4 ⊢ 𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21		fvex 6787	. . . . 5 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ V
22	21	rankss 9607	. . . 4 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
23	20, 22	ax-mp 5	. . 3 ⊢ (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
24		r1ord3 9540	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
25	14, 24	mpan 687	. . . . . 6 ⊢ (𝑦 ∈ On → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
26	25	ss2rabi 4010	. . . . 5 ⊢ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
27		intss 4900	. . . . 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})
28	26, 27	ax-mp 5	. . . 4 ⊢ ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)} ⊆ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
29		rankval2 9576	. . . . 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)})
30	21, 29	ax-mp 5	. . . 4 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) = ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
31		intmin 4899	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On → ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
32	14, 31	ax-mp 5	. . . . 5 ⊢ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
33	32	eqcomi 2747	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
34	28, 30, 33	3sstr4i 3964	. . 3 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
35	23, 34	sstri 3930	. 2 ⊢ (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
36		iunss 4975	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
37	11	rankel 9597	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38		rankon 9553	. . . . 5 ⊢ (rank‘𝐴) ∈ On
39	9, 38	onsucssi 7688	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
40	37, 39	sylib 217	. . 3 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
41	36, 40	mprgbir 3079	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
42	35, 41	eqssi 3937	1 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∩ cint 4879  ∪ ciun 4924  Oncon0 6266  suc csuc 6268  ‘cfv 6433  𝑅₁cr1 9520  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by:  rankbnd  9626  rankc1  9628  scottrankd  41866