Theorem karden 9964

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 10620). 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 9963 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 5169	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 9045	. . . . . . 7 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3545	. . . . . 6 ⊢ ∃𝑤 𝑤 ≈ 𝐴
5		abn0 4408	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ ∃𝑤 𝑤 ≈ 𝐴)
6	4, 5	mpbir 231	. . . . 5 ⊢ {𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅
7		scott0 9955	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} = ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
8	7	necon3bii 2999	. . . . 5 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
9	6, 8	mpbi 230	. . . 4 ⊢ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
10		rabn0 4412	. . . 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 230	. . 3 ⊢ ∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
12		vex 3492	. . . . . . . 8 ⊢ 𝑧 ∈ V
13		breq1 5169	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
14	12, 13	elab 3694	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ↔ 𝑧 ≈ 𝐴)
15		breq1 5169	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
16	15	ralab 3713	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
17	14, 16	anbi12i 627	. . . . . 6 ⊢ ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18		simpl 482	. . . . . . . . 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 2758	. . . . . . . . . . 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 2814	. . . . . . . . . . 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 275	. . . . . . . . . 10 ⊢ (𝐶 = 𝐷 ↔ ∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25		breq1 5169	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
26		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
27	26	sseq1d 4040	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
28	27	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
29	28	albidv 1919	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
30	25, 29	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31		breq1 5169	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐵 ↔ 𝑧 ≈ 𝐵))
32	27	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
33	32	albidv 1919	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
34	31, 33	anbi12d 631	. . . . . . . . . . . 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 345	. . . . . . . . . . 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 1996	. . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
38		simpl 482	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵)
39	37, 38	biimtrdi 253	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵))
40	19, 39	jcad 512	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵)))
41		ensym 9063	. . . . . . . 8 ⊢ (𝑧 ≈ 𝐴 → 𝐴 ≈ 𝑧)
42		entr 9066	. . . . . . . 8 ⊢ ((𝐴 ≈ 𝑧 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
43	41, 42	sylan 579	. . . . . . 7 ⊢ ((𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
44	40, 43	syl6 35	. . . . . 6 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴 ≈ 𝐵))
45	17, 44	biimtrid 242	. . . . 5 ⊢ (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴 ≈ 𝐵))
46	45	expd 415	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} → (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵)))
47	46	rexlimdv 3159	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴}∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵))
48	11, 47	mpi 20	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
49		enen2 9184	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
50		enen2 9184	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
51	50	imbi1d 341	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
52	51	albidv 1919	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
53	49, 52	anbi12d 631	. . . 4 ⊢ (𝐴 ≈ 𝐵 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
54	53	abbidv 2811	. . 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 2805	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐶 = 𝐷)
56	48, 55	impbii 209	1 ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  ‘cfv 6573   ≈ cen 9000  rankcrnk 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-r1 9833  df-rank 9834

This theorem is referenced by: (None)