Theorem rankse 35823

Description: Construct a relation that is set-like on any class. (Contributed by BTernaryTau, 18-Sep-2026.)

Hypothesis

Ref	Expression
rankse.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}

Assertion

Ref	Expression
rankse	⊢ 𝑆 Se 𝐴

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑆(𝑥, 𝑦)

Proof of Theorem rankse

Dummy variables 𝑤 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankon 9830	. . . . . 6 ⊢ (rank‘𝑦) ∈ On
2		id 23	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦))
3	2	rgenw 3084	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 ((rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦))
4		fveq2 6908	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → (rank‘𝑣) = (rank‘𝑥))
5	4	sseq1d 3944	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → ((rank‘𝑣) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
6	5	ralrab2 3659	. . . . . . 7 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ (rank‘𝑦) ↔ ∀𝑥 ∈ 𝐴 ((rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦)))
7	3, 6	mpbir 234	. . . . . 6 ⊢ ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ (rank‘𝑦)
8		sseq2 3939	. . . . . . . 8 ⊢ (𝑢 = (rank‘𝑦) → ((rank‘𝑣) ⊆ 𝑢 ↔ (rank‘𝑣) ⊆ (rank‘𝑦)))
9	8	ralbidv 3189	. . . . . . 7 ⊢ (𝑢 = (rank‘𝑦) → (∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ 𝑢 ↔ ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ (rank‘𝑦)))
10	9	rspcev 3580	. . . . . 6 ⊢ (((rank‘𝑦) ∈ On ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ (rank‘𝑦)) → ∃𝑢 ∈ On ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ 𝑢)
11	1, 7, 10	mp2an 705	. . . . 5 ⊢ ∃𝑢 ∈ On ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ 𝑢
12		bndrank 9881	. . . . 5 ⊢ (∃𝑢 ∈ On ∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} (rank‘𝑣) ⊆ 𝑢 → {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V)
13	11, 12	ax-mp 5	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
14		simpl 488	. . . . . . . 8 ⊢ (((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑) → (rank‘𝑥) = (rank‘𝑦))
15	14	orim2i 924	. . . . . . 7 ⊢ (((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑)) → ((rank‘𝑥) ∈ (rank‘𝑦) ∨ (rank‘𝑥) = (rank‘𝑦)))
16		rankse.1	. . . . . . . . 9 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}
17	16	breqi 5112	. . . . . . . 8 ⊢ (𝑥𝑆𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}𝑦)
18		df-br 5107	. . . . . . . 8 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))})
19		opabidw 5502	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))} ↔ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑)))
20	17, 18, 19	3bitri 300	. . . . . . 7 ⊢ (𝑥𝑆𝑦 ↔ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑)))
21		rankon 9830	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
22	21, 1	onsseli 6503	. . . . . . 7 ⊢ ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ (rank‘𝑥) = (rank‘𝑦)))
23	15, 20, 22	3imtr4i 295	. . . . . 6 ⊢ (𝑥𝑆𝑦 → (rank‘𝑥) ⊆ (rank‘𝑦))
24	23	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥𝑆𝑦 → (rank‘𝑥) ⊆ (rank‘𝑦)))
25	24	ss2rabi 4004	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)}
26	13, 25	ssexi 5288	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
27	26	rgenw 3084	. 2 ⊢ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
28		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥𝐴
29		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑧𝐴
30		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑧 𝑥𝑆𝑤
31		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
32		nfopab1 5178	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}
33	16, 32	nfcxfr 2924	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
34		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
35	31, 33, 34	nfbr 5155	. . . . . 6 ⊢ Ⅎ𝑥 𝑧𝑆𝑤
36		breq1 5109	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝑆𝑤 ↔ 𝑧𝑆𝑤))
37	28, 29, 30, 35, 36	cbvrabw 3450	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} = {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤}
38	37	eleq1i 2855	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V ↔ {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤} ∈ V)
39	38	ralbii 3112	. . 3 ⊢ (∀𝑤 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V ↔ ∀𝑤 ∈ 𝐴 {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤} ∈ V)
40		nfv 1949	. . . 4 ⊢ Ⅎ𝑤{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
41		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝑥
42		nfopab2 5179	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}
43	16, 42	nfcxfr 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑆
44		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
45	41, 43, 44	nfbr 5155	. . . . . 6 ⊢ Ⅎ𝑦 𝑥𝑆𝑤
46		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑦𝐴
47	45, 46	nfrabw 3451	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤}
48	47	nfel1 2942	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V
49		breq2 5110	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥𝑆𝑦 ↔ 𝑥𝑆𝑤))
50	49	rabbidv 3423	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤})
51	50	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V))
52	40, 48, 51	cbvralw 3308	. . 3 ⊢ (∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V ↔ ∀𝑤 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V)
53		df-se 5609	. . 3 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑤 ∈ 𝐴 {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤} ∈ V)
54	39, 52, 53	3bitr4ri 307	. 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V)
55	27, 54	mpbir 234	1 ⊢ 𝑆 Se 𝐴

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106  {copab 5170   Se wse 5606  Oncon0 6380  ‘cfv 6558  rankcrnk 9794

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613  ax-inf2 9669

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-rank 9796

This theorem is used by:  wevsev  36041