Theorem rankval4b 35408

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 9827 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 35404	. . . 4 ⊢ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ∈ ∪ (𝑅1 “ On)
2		rankon 9755	. . . . . . . . . . 11 ⊢ (rank‘𝑥) ∈ On
3	2	onsuci 7823	. . . . . . . . . 10 ⊢ suc (rank‘𝑥) ∈ On
4	3	rgenw 3083	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On
5		iunon 8314	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
6	4, 5	mpan2 703	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ∈ On)
7		r1ord3 9742	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
9		ssiun2 5008	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
10	8, 9	impel 514	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
11		elwf 35405	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
12		rankidb 9760	. . . . . . . . 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 3940	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)))
15	14	ex 417	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
16	15	alrimiv 1950	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))))
17		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑅1
19		nfiu1 4988	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)
20	18, 19	nffv 6881	. . . . . 6 ⊢ Ⅎ𝑥(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
21	17, 20	dfssf 3930	. . . . 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 9808	. . . 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 9742	. . . . . . 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 591	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ On) → (∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)))
27	26	ss2rabdv 4031	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥)) ⊆ (𝑅1‘𝑦)})
28		intss 4930	. . . . 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 35407	. . . . 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 4929	. . . . . 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 2771	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝑦})
35	29, 31, 34	3sstr4d 3994	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘(𝑅1‘∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
36	24, 35	sstrd 3949	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
37		rankelb 9784	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
38		rankon 9755	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
39	2, 38	onsucssi 7825	. . . . 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 3156	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
42		iunss 5005	. . 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 3956	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1561   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417   ⊆ wss 3907  ∪ cuni 4868  ∩ cint 4908  ∪ ciun 4952   “ cima 5655  Oncon0 6350  suc csuc 6352  ‘cfv 6525  𝑅₁cr1 9722  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-r1 9724  df-rank 9725

This theorem is referenced by:  rankfilimbi  35409