Theorem rankval4 9904

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 2902	. . . . . 6 ⊢ Ⅎ𝑥𝐴
2		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
3		nfiu1 5031	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
4	2, 3	nffv 6916	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
5	1, 4	dfssf 3985	. . . . 5 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
6		vex 3481	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankid 9870	. . . . . 6 ⊢ 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8		ssiun2 5051	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
9		rankon 9832	. . . . . . . . . 10 ⊢ (rank‘𝑥) ∈ On
10	9	onsuci 7858	. . . . . . . . 9 ⊢ suc (rank‘𝑥) ∈ On
11		rankr1b.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
12	10	rgenw 3062	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
13		iunon 8377	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
14	11, 12, 13	mp2an 692	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
15		r1ord3 9819	. . . . . . . . 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 692	. . . . . . . 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 3993	. . . . . 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 1795	. . . 4 ⊢ 𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21		fvex 6919	. . . . 5 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ V
22	21	rankss 9886	. . . 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 9819	. . . . . . 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 690	. . . . . 6 ⊢ (𝑦 ∈ On → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
26	25	ss2rabi 4086	. . . . 5 ⊢ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)}
27		intss 4973	. . . . 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 9855	. . . . 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 4972	. . . . . 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 2743	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦}
34	28, 30, 33	3sstr4i 4038	. . 3 ⊢ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
35	23, 34	sstri 4004	. 2 ⊢ (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
36		iunss 5049	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
37	11	rankel 9876	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38		rankon 9832	. . . . 5 ⊢ (rank‘𝐴) ∈ On
39	9, 38	onsucssi 7861	. . . 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 3065	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
42	35, 41	eqssi 4011	1 ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  ∀wral 3058  {crab 3432  Vcvv 3477   ⊆ wss 3962  ∩ cint 4950  ∪ ciun 4995  Oncon0 6385  suc csuc 6387  ‘cfv 6562  𝑅₁cr1 9799  rankcrnk 9800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-r1 9801  df-rank 9802

This theorem is referenced by:  rankbnd  9905  rankc1  9907  scottrankd  44243