Theorem karden 9869

Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10523). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9868 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)

Hypotheses

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

Assertion

Ref	Expression
karden	⊢ (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 karden

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

Step	Hyp	Ref	Expression
1		karden.a	. . . . . . 7 ⊢ 𝐴 ∈ V
2		breq1 5107	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8970	. . . . . . 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		scott0 9848	. . . . . 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 5107	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
14	12, 13	elab 3641	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ↔ 𝑧 ≈ 𝐴)
15		breq1 5107	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
16	15	ralab 3659	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
17	14, 16	anbi12i 639	. . . . . 6 ⊢ ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18		simpl 487	. . . . . . . . 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		karden.c	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21		karden.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 5107	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
26		fveq2 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
27	26	sseq1d 3970	. . . . . . . . . . . . . . 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 1943	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
30	25, 29	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31		breq1 5107	. . . . . . . . . . . . 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 1943	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
34	31, 33	anbi12d 643	. . . . . . . . . . . 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 2011	. . . . . . . . . 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 487	. . . . . . . . 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 521	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵)))
41		ensym 8988	. . . . . . . 8 ⊢ (𝑧 ≈ 𝐴 → 𝐴 ≈ 𝑧)
42		entr 8991	. . . . . . . 8 ⊢ ((𝐴 ≈ 𝑧 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
43	41, 42	sylan 591	. . . . . . 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 420	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} → (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵)))
47	46	rexlimdv 3164	. . 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 9094	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
50		enen2 9094	. . . . . . 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 1943	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
53	49, 52	anbi12d 643	. . . 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288   class class class wbr 5104  ‘cfv 6525   ≈ cen 8928  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-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  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 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  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-er 8682  df-en 8932  df-r1 9724  df-rank 9725

This theorem is referenced by: (None)