Theorem kardenOLD 9896

Description: Obsolete version of karden 9895 as of 19-Jul-2026. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
kardenOLD.a	⊢ 𝐴 ∈ V
kardenOLD.c	⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
kardenOLD.d	⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Assertion

Ref	Expression
kardenOLD	⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐷(𝑥, 𝑦)

Proof of Theorem kardenOLD

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

Step	Hyp	Ref	Expression
1		kardenOLD.a	. . . . . . 7 ⊢ 𝐴 ∈ V
2		breq1 5114	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8988	. . . . . . 7 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3506	. . . . . 6 ⊢ ∃𝑤 𝑤 ≈ 𝐴
5		abn0 4341	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ ∃𝑤 𝑤 ≈ 𝐴)
6	4, 5	mpbir 234	. . . . 5 ⊢ {𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅
7		scott0OLD 9874	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} = ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
8	7	necon3bii 3012	. . . . 5 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
9	6, 8	mpbi 233	. . . 4 ⊢ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
10		rabn0 4346	. . . 4 ⊢ ({𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦))
11	9, 10	mpbi 233	. . 3 ⊢ ∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
12		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
13		breq1 5114	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
14	12, 13	elab 3640	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ↔ 𝑧 ≈ 𝐴)
15		breq1 5114	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
16	15	ralab 3658	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
17	14, 16	anbi12i 640	. . . . . 6 ⊢ ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18		simpl 488	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐴)
19	18	a1i 11	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐴))
20		kardenOLD.c	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21		kardenOLD.d	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
22	20, 21	eqeq12i 2783	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
23		abbib 2834	. . . . . . . . . . 11 ⊢ ({𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ ∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
24	22, 23	bitri 278	. . . . . . . . . 10 ⊢ (𝐶 = 𝐷 ↔ ∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25		breq1 5114	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
26		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
27	26	sseq1d 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
28	27	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
29	28	albidv 1953	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
30	25, 29	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31		breq1 5114	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐵 ↔ 𝑧 ≈ 𝐵))
32	27	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
33	32	albidv 1953	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
34	31, 33	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
35	30, 34	bibi12d 348	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))))
36	35	spvv 2021	. . . . . . . . . 10 ⊢ (∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
37	24, 36	sylbi 220	. . . . . . . . 9 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
38		simpl 488	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵)
39	37, 38	biimtrdi 256	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵))
40	19, 39	jcad 522	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵)))
41		ensym 9006	. . . . . . . 8 ⊢ (𝑧 ≈ 𝐴 → 𝐴 ≈ 𝑧)
42		entr 9009	. . . . . . . 8 ⊢ ((𝐴 ≈ 𝑧 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
43	41, 42	sylan 592	. . . . . . 7 ⊢ ((𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
44	40, 43	syl6 36	. . . . . 6 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴 ≈ 𝐵))
45	17, 44	biimtrid 245	. . . . 5 ⊢ (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴 ≈ 𝐵))
46	45	expd 421	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} → (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵)))
47	46	rexlimdv 3166	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵))
48	11, 47	mpi 21	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
49		enen2 9113	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
50		enen2 9113	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
51	50	imbi1d 344	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
52	51	albidv 1953	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
53	49, 52	anbi12d 644	. . . 4 ⊢ (𝐴 ≈ 𝐵 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
54	53	abbidv 2831	. . 3 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
55	54, 20, 21	3eqtr4g 2825	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐶 = 𝐷)
56	48, 55	impbii 212	1 ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ‘cfv 6540   ≈ cen 8946  rankcrnk 9742

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-r1 9743  df-rank 9744

This theorem is used by: (None)