Theorem rankval4b 35615

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. This variant of rankval4 9853 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅₁. (Contributed by BTernaryTau, 19-Jan-2026.)

Assertion

Ref	Expression
rankval4b	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval4b

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1wf 35611	. . . 4 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ ∪ (𝑅1 “ On)
2		rankon 9781	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ On
3	2	onsuci 7839	. . . . . . . . . 10 ⊢ suc (rank‘𝑥) ∈ On
4	3	rgenw 3082	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
5		iunon 8332	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
6	4, 5	mpan2 704	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
7		r1ord3 9768	. . . . . . . . . 10 ⊢ ((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))))
8	3, 6, 7	sylancr 599	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
9		ssiun2 5010	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
10	8, 9	impel 515	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
11		elwf 35612	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
12		rankidb 9786	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ∈ (𝑅1‘suc (rank‘𝑥)))
13	11, 12	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘suc (rank‘𝑥)))
14	10, 13	sseldd 3935	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
15	14	ex 418	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
16	15	alrimiv 1960	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
17		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
19		nfiu1 4990	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
20	18, 19	nffv 6892	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21	17, 20	dfssf 3925	. . . . 5 ⊢ (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
22	16, 21	sylibr 237	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
23		rankssb 9834	. . . 4 ⊢ ((𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))))
24	1, 22, 23	mpsyl 69	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
25		r1ord3 9768	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
26	6, 25	sylan 592	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ On) → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
27	26	ss2rabdv 4026	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)})
28		intss 4932	. . . . 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})
29	27, 28	syl 18	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)} ⊆ ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦})
30		rankval2b 35614	. . . . 5 ⊢ ((𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ ∪ (𝑅1 “ On) → (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) = ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)})
31	1, 30	mp1i 14	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) = ∩ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)})
32		intmin 4931	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On → ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
33	6, 32	syl 18	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
34	33	eqcomd 2768	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦})
35	29, 31, 34	3sstr4d 3989	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
36	24, 35	sstrd 3944	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
37		rankelb 9810	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
38		rankon 9781	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
39	2, 38	onsucssi 7841	. . . . 5 ⊢ ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
40	37, 39	imbitrdi 254	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴)))
41	40	ralrimiv 3155	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
42		iunss 5007	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
43	41, 42	sylibr 237	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
44	36, 43	eqssd 3951	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ⊆ wss 3902  ∪ cuni 4870  ∩ cint 4910  ∪ ciun 4954   “ cima 5662  Oncon0 6361  suc csuc 6363  ‘cfv 6537  𝑅₁cr1 9748  rankcrnk 9749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-r1 9750  df-rank 9751

This theorem is used by:  rankfilimbi  35617