Theorem rankse 35786

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 9804	. . . . . 6 ⊢ (rank‘𝑦) ∈ On
2		id 23	. . . . . . . 8 ⊢ ((rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦))
3	2	rgenw 3080	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 ((rank‘𝑥) ⊆ (rank‘𝑦) → (rank‘𝑥) ⊆ (rank‘𝑦))
4		fveq2 6885	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → (rank‘𝑣) = (rank‘𝑥))
5	4	sseq1d 3939	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → ((rank‘𝑣) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
6	5	ralrab2 3655	. . . . . . 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 3934	. . . . . . . 8 ⊢ (𝑢 = (rank‘𝑦) → ((rank‘𝑣) ⊆ 𝑢 ↔ (rank‘𝑣) ⊆ (rank‘𝑦)))
9	8	ralbidv 3185	. . . . . . 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 3576	. . . . . 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 9855	. . . . 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 5108	. . . . . . . 8 ⊢ (𝑥𝑆𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}𝑦)
18		df-br 5103	. . . . . . . 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 5497	. . . . . . . 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 9804	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
22	21, 1	onsseli 6485	. . . . . . 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 4001	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ⊆ {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) ⊆ (rank‘𝑦)}
26	13, 25	ssexi 5283	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
27	26	rgenw 3080	. 2 ⊢ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
28		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝐴
29		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑧𝐴
30		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑧 𝑥𝑆𝑤
31		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
32		nfopab1 5174	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}
33	16, 32	nfcxfr 2920	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
34		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
35	31, 33, 34	nfbr 5151	. . . . . 6 ⊢ Ⅎ𝑥 𝑧𝑆𝑤
36		breq1 5105	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝑆𝑤 ↔ 𝑧𝑆𝑤))
37	28, 29, 30, 35, 36	cbvrabw 3446	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} = {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤}
38	37	eleq1i 2851	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V ↔ {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤} ∈ V)
39	38	ralbii 3108	. . 3 ⊢ (∀𝑤 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V ↔ ∀𝑤 ∈ 𝐴 {𝑧 ∈ 𝐴 ∣ 𝑧𝑆𝑤} ∈ V)
40		nfv 1946	. . . 4 ⊢ Ⅎ𝑤{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V
41		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦𝑥
42		nfopab2 5175	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}
43	16, 42	nfcxfr 2920	. . . . . . 7 ⊢ Ⅎ𝑦𝑆
44		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
45	41, 43, 44	nfbr 5151	. . . . . 6 ⊢ Ⅎ𝑦 𝑥𝑆𝑤
46		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑦𝐴
47	45, 46	nfrabw 3447	. . . . 5 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤}
48	47	nfel1 2938	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V
49		breq2 5106	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑥𝑆𝑦 ↔ 𝑥𝑆𝑤))
50	49	rabbidv 3419	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤})
51	50	eleq1d 2845	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V))
52	40, 48, 51	cbvralw 3304	. . 3 ⊢ (∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑦} ∈ V ↔ ∀𝑤 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑆𝑤} ∈ V)
53		df-se 5604	. . 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ⟨cop 4589   class class class wbr 5102  {copab 5166   Se wse 5601  Oncon0 6362  ‘cfv 6538  rankcrnk 9768

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9587  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-r1 9769  df-rank 9770

This theorem is used by:  wevsev  35953