Theorem karden 8641

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 9252). 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 8640 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.)

Hypotheses

Ref	Expression
karden.1	⊢ 𝐴 ∈ V
karden.2	⊢ 𝐵 ∈ V
karden.3	⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
karden.4	⊢ 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.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
2	1	enref 7874	. . . . . . 7 ⊢ 𝐴 ≈ 𝐴
3		breq1 4586	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
4	1, 3	spcev 3273	. . . . . . 7 ⊢ (𝐴 ≈ 𝐴 → ∃𝑤 𝑤 ≈ 𝐴)
5	2, 4	ax-mp 5	. . . . . 6 ⊢ ∃𝑤 𝑤 ≈ 𝐴
6		abn0 3908	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ ∃𝑤 𝑤 ≈ 𝐴)
7	5, 6	mpbir 220	. . . . 5 ⊢ {𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅
8		scott0 8632	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} = ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
9	8	necon3bii 2834	. . . . 5 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
10	7, 9	mpbi 219	. . . 4 ⊢ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
11		rabn0 3912	. . . 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))
12	10, 11	mpbi 219	. . 3 ⊢ ∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
13		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
14		breq1 4586	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
15	13, 14	elab 3319	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ↔ 𝑧 ≈ 𝐴)
16		breq1 4586	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	ralab 3334	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
18	15, 17	anbi12i 729	. . . . . 6 ⊢ ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
19		simpl 472	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐴)
20	19	a1i 11	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐴))
21		karden.3	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
22		karden.4	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
23	21, 22	eqeq12i 2624	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
24		abbi 2724	. . . . . . . . . . 11 ⊢ (∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
25	23, 24	bitr4i 266	. . . . . . . . . 10 ⊢ (𝐶 = 𝐷 ↔ ∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
26		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
27		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
28	27	sseq1d 3595	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
29	28	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
30	29	albidv 1836	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
31	26, 30	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
32		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐵 ↔ 𝑧 ≈ 𝐵))
33	28	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
34	33	albidv 1836	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
35	32, 34	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
36	31, 35	bibi12d 334	. . . . . . . . . . 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))))))
37	36	spv 2248	. . . . . . . . . 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)))))
38	25, 37	sylbi 206	. . . . . . . . 9 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
39		simpl 472	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵)
40	38, 39	syl6bi 242	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵))
41	20, 40	jcad 554	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵)))
42		ensym 7891	. . . . . . . 8 ⊢ (𝑧 ≈ 𝐴 → 𝐴 ≈ 𝑧)
43		entr 7894	. . . . . . . 8 ⊢ ((𝐴 ≈ 𝑧 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
44	42, 43	sylan 487	. . . . . . 7 ⊢ ((𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
45	41, 44	syl6 34	. . . . . 6 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴 ≈ 𝐵))
46	18, 45	syl5bi 231	. . . . 5 ⊢ (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴 ≈ 𝐵))
47	46	expd 451	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} → (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵)))
48	47	rexlimdv 3012	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵))
49	12, 48	mpi 20	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
50		enen2 7986	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
51		enen2 7986	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
52	51	imbi1d 330	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
53	52	albidv 1836	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
54	50, 53	anbi12d 743	. . . 4 ⊢ (𝐴 ≈ 𝐵 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
55	54	abbidv 2728	. . 3 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
56	55, 21, 22	3eqtr4g 2669	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐶 = 𝐷)
57	49, 56	impbii 198	1 ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804   ≈ cen 7838  rankcrnk 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-r1 8510  df-rank 8511

This theorem is referenced by: (None)