Theorem karden 9902

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 10563). 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 9900 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.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)

Hypotheses

Ref	Expression
karden.a	⊢ 𝐴 ∈ V
karden.c	⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
karden.d	⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem karden

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		karden.a	. . . . . 6 ⊢ 𝐴 ∈ V
2		breq1 5110	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8995	. . . . . 6 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3502	. . . . 5 ⊢ ∃𝑥 𝑥 ≈ 𝐴
5		karden.c	. . . . . . 7 ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
6	5	neeq1i 3021	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
7		scott0b 9880	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = ∅)
8	7	necon3bii 3009	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
9		abn0 4337	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ ∃𝑥 𝑥 ≈ 𝐴)
10	6, 8, 9	3bitr2i 302	. . . . 5 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ≈ 𝐴)
11	4, 10	mpbir 234	. . . 4 ⊢ 𝐶 ≠ ∅
12		n0 4303	. . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐶)
13	11, 12	mpbi 233	. . 3 ⊢ ∃𝑦 𝑦 ∈ 𝐶
14		eleq2 2851	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
15	14	pm4.71da 573	. . . . 5 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		breq1 5110	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	elscottab 9885	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} → 𝑦 ≈ 𝐴)
18	17, 5	eleq2s 2880	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ≈ 𝐴)
19	18	ensymd 9015	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝑦)
20		breq1 5110	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐵 ↔ 𝑦 ≈ 𝐵))
21	20	elscottab 9885	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐵} → 𝑦 ≈ 𝐵)
22		karden.d	. . . . . . 7 ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}
23	21, 22	eleq2s 2880	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≈ 𝐵)
24		entr 9016	. . . . . 6 ⊢ ((𝐴 ≈ 𝑦 ∧ 𝑦 ≈ 𝐵) → 𝐴 ≈ 𝐵)
25	19, 23, 24	syl2an 608	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝐴 ≈ 𝐵)
26	15, 25	biimtrdi 256	. . . 4 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝐵))
27	26	exlimdv 1966	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑦 𝑦 ∈ 𝐶 → 𝐴 ≈ 𝐵))
28	13, 27	mpi 21	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
29		enen2 9120	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
30	29	abbidv 2828	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ 𝑥 ≈ 𝐴} = {𝑥 ∣ 𝑥 ≈ 𝐵})
31	30	scotteqd 9873	. . 3 ⊢ (𝐴 ≈ 𝐵 → Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = Scott {𝑥 ∣ 𝑥 ≈ 𝐵})
32	31, 5, 22	3eqtr4g 2822	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐶 = 𝐷)
33	28, 32	impbii 212	1 ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740   ≠ wne 2957  Vcvv 3453  ∅c0 4282   class class class wbr 5107   ≈ cen 8953  Scott cscott 9871

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8957  df-r1 9750  df-rank 9751  df-scott 9872

This theorem is used by:  kardeng  35691