Theorem karden 9476

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 10130). 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 9475 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 5042	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑤 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8639	. . . . . . 7 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3447	. . . . . 6 ⊢ ∃𝑤 𝑤 ≈ 𝐴
5		abn0 4281	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅ ↔ ∃𝑤 𝑤 ≈ 𝐴)
6	4, 5	mpbir 234	. . . . 5 ⊢ {𝑤 ∣ 𝑤 ≈ 𝐴} ≠ ∅
7		scott0 9467	. . . . . 6 ⊢ ({𝑤 ∣ 𝑤 ≈ 𝐴} = ∅ ↔ {𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
8	7	necon3bii 2984	. . . . 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 4286	. . . 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 3402	. . . . . . . 8 ⊢ 𝑧 ∈ V
13		breq1 5042	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
14	12, 13	elab 3576	. . . . . . 7 ⊢ (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ↔ 𝑧 ≈ 𝐴)
15		breq1 5042	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
16	15	ralab 3595	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
17	14, 16	anbi12i 630	. . . . . 6 ⊢ ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18		simpl 486	. . . . . . . . 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 2751	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
23		abbi 2803	. . . . . . . . . . 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)))})
24	22, 23	bitr4i 281	. . . . . . . . . 10 ⊢ (𝐶 = 𝐷 ↔ ∀𝑥((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25		breq1 5042	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐴 ↔ 𝑧 ≈ 𝐴))
26		fveq2 6695	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
27	26	sseq1d 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
28	27	imbi2d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
29	28	albidv 1928	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
30	25, 29	anbi12d 634	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31		breq1 5042	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ≈ 𝐵 ↔ 𝑧 ≈ 𝐵))
32	27	imbi2d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
33	32	albidv 1928	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
34	31, 33	anbi12d 634	. . . . . . . . . . . 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 349	. . . . . . . . . . 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 2006	. . . . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝑧 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵)
39	37, 38	syl6bi 256	. . . . . . . 8 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧 ≈ 𝐵))
40	19, 39	jcad 516	. . . . . . 7 ⊢ (𝐶 = 𝐷 → ((𝑧 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧 ≈ 𝐴 ∧ 𝑧 ≈ 𝐵)))
41		ensym 8655	. . . . . . . 8 ⊢ (𝑧 ≈ 𝐴 → 𝐴 ≈ 𝑧)
42		entr 8658	. . . . . . . 8 ⊢ ((𝐴 ≈ 𝑧 ∧ 𝑧 ≈ 𝐵) → 𝐴 ≈ 𝐵)
43	41, 42	sylan 583	. . . . . . 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	syl5bi 245	. . . . 5 ⊢ (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴 ≈ 𝐵))
46	45	expd 419	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑧 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} → (∀𝑦 ∈ {𝑤 ∣ 𝑤 ≈ 𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴 ≈ 𝐵)))
47	46	rexlimdv 3192	. . 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 8765	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
50		enen2 8765	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ 𝐵))
51	50	imbi1d 345	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ((𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
52	51	albidv 1928	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
53	49, 52	anbi12d 634	. . . 4 ⊢ (𝐴 ≈ 𝐵 → ((𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
54	53	abbidv 2800	. . 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 2796	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐶 = 𝐷)
56	48, 55	impbii 212	1 ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3052  {crab 3055  Vcvv 3398   ⊆ wss 3853  ∅c0 4223   class class class wbr 5039  ‘cfv 6358   ≈ cen 8601  rankcrnk 9344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-iin 4893  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-r1 9345  df-rank 9346

This theorem is referenced by: (None)