Theorem rankval4 9781

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 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐴
2		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
3		nfiu1 4981	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
4	2, 3	nffv 6843	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
5	1, 4	dfssf 3923	. . . . 5 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
6		vex 3443	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankid 9747	. . . . . 6 ⊢ 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8		ssiun2 5002	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
9		rankon 9709	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
10	9	onsuci 7781	. . . . . . . . 9 ⊢ suc (rank‘𝑥) ∈ On
11		rankr1b.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
12	10	rgenw 3054	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
13		iunon 8271	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
14	11, 12, 13	mp2an 693	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
15		r1ord3 9696	. . . . . . . . 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 693	. . . . . . . 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 3931	. . . . . 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 1801	. . . 4 ⊢ 𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21		fvex 6846	. . . . 5 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ V
22	21	rankss 9763	. . . 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 9696	. . . . . . 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 691	. . . . . 6 ⊢ (𝑦 ∈ On → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
26	25	ss2rabi 4027	. . . . 5 ⊢ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
27		intss 4923	. . . . 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 9732	. . . . 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 4922	. . . . . 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 2744	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
34	28, 30, 33	3sstr4i 3984	. . 3 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
35	23, 34	sstri 3942	. 2 ⊢ (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
36		iunss 4999	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
37	11	rankel 9753	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38		rankon 9709	. . . . 5 ⊢ (rank‘𝐴) ∈ On
39	9, 38	onsucssi 7783	. . . 4 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
40	37, 39	sylib 218	. . 3 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
41	36, 40	mprgbir 3057	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
42	35, 41	eqssi 3949	1 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3050  {crab 3398  Vcvv 3439   ⊆ wss 3900  ∩ cint 4901  ∪ ciun 4945  Oncon0 6316  suc csuc 6318  ‘cfv 6491  𝑅₁cr1 9676  rankcrnk 9677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-reg 9499  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9678  df-rank 9679

This theorem is referenced by:  rankbnd  9782  rankc1  9784  scottrankd  44526