Theorem karden 9898

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 10553). 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 9896 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 5117	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 8991	. . . . . 6 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3507	. . . . 5 ⊢ ∃𝑥 𝑥 ≈ 𝐴
5		karden.c	. . . . . . 7 ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
6	5	neeq1i 3025	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
7		scott0b 9876	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = ∅)
8	7	necon3bii 3013	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
9		abn0 4344	. . . . . 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 4310	. . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐶)
13	11, 12	mpbi 233	. . 3 ⊢ ∃𝑦 𝑦 ∈ 𝐶
14		eleq2 2855	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
15	14	pm4.71da 573	. . . . 5 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		breq1 5117	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	elscottab 9881	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} → 𝑦 ≈ 𝐴)
18	17, 5	eleq2s 2884	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ≈ 𝐴)
19	18	ensymd 9011	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝑦)
20		breq1 5117	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐵 ↔ 𝑦 ≈ 𝐵))
21	20	elscottab 9881	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐵} → 𝑦 ≈ 𝐵)
22		karden.d	. . . . . . 7 ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}
23	21, 22	eleq2s 2884	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≈ 𝐵)
24		entr 9012	. . . . . 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 9116	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
30	29	abbidv 2832	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ 𝑥 ≈ 𝐴} = {𝑥 ∣ 𝑥 ≈ 𝐵})
31	30	scotteqd 9869	. . 3 ⊢ (𝐴 ≈ 𝐵 → Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = Scott {𝑥 ∣ 𝑥 ≈ 𝐵})
32	31, 5, 22	3eqtr4g 2826	. 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 2146  {cab 2744   ≠ wne 2961  Vcvv 3458  ∅c0 4289   class class class wbr 5114   ≈ cen 8949  Scott cscott 9867

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-r1 9746  df-rank 9747  df-scott 9868

This theorem is used by:  kardeng  35594