Theorem karden 9940

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 10616). 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 9938 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 5106	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8996	. . . . . 6 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3500	. . . . 5 ⊢ ∃𝑥 𝑥 ≈ 𝐴
5		karden.c	. . . . . . 7 ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
6	5	neeq1i 3020	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
7		scott0b 9918	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = ∅)
8	7	necon3bii 3008	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
9		abn0 4334	. . . . . 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 4300	. . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐶)
13	11, 12	mpbi 233	. . 3 ⊢ ∃𝑦 𝑦 ∈ 𝐶
14		eleq2 2850	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
15	14	pm4.71da 573	. . . . 5 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	elscottab 9923	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} → 𝑦 ≈ 𝐴)
18	17, 5	eleq2s 2879	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ≈ 𝐴)
19	18	ensymd 9016	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝑦)
20		breq1 5106	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐵 ↔ 𝑦 ≈ 𝐵))
21	20	elscottab 9923	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐵} → 𝑦 ≈ 𝐵)
22		karden.d	. . . . . . 7 ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}
23	21, 22	eleq2s 2879	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≈ 𝐵)
24		entr 9017	. . . . . 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 9121	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
30	29	abbidv 2827	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ 𝑥 ≈ 𝐴} = {𝑥 ∣ 𝑥 ≈ 𝐵})
31	30	scotteqd 9911	. . 3 ⊢ (𝐴 ≈ 𝐵 → Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = Scott {𝑥 ∣ 𝑥 ≈ 𝐵})
32	31, 5, 22	3eqtr4g 2821	. 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 2739   ≠ wne 2956  Vcvv 3451  ∅c0 4279   class class class wbr 5103   ≈ cen 8954  Scott cscott 9909

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-r1 9752  df-rank 9753  df-scott 9910

This theorem is used by:  kardeng  35798