Theorem karden 9962

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 10638). 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 9960 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 5105	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 9015	. . . . . 6 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3499	. . . . 5 ⊢ ∃𝑥 𝑥 ≈ 𝐴
5		karden.c	. . . . . . 7 ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
6	5	neeq1i 3019	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
7		scott0b 9940	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = ∅)
8	7	necon3bii 3007	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
9		abn0 4333	. . . . . 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 4299	. . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐶)
13	11, 12	mpbi 233	. . 3 ⊢ ∃𝑦 𝑦 ∈ 𝐶
14		eleq2 2849	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
15	14	pm4.71da 573	. . . . 5 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	elscottab 9945	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} → 𝑦 ≈ 𝐴)
18	17, 5	eleq2s 2878	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ≈ 𝐴)
19	18	ensymd 9035	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝑦)
20		breq1 5105	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐵 ↔ 𝑦 ≈ 𝐵))
21	20	elscottab 9945	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐵} → 𝑦 ≈ 𝐵)
22		karden.d	. . . . . . 7 ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}
23	21, 22	eleq2s 2878	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≈ 𝐵)
24		entr 9036	. . . . . 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 1965	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑦 𝑦 ∈ 𝐶 → 𝐴 ≈ 𝐵))
28	13, 27	mpi 21	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
29		enen2 9140	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
30	29	abbidv 2826	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ 𝑥 ≈ 𝐴} = {𝑥 ∣ 𝑥 ≈ 𝐵})
31	30	scotteqd 9933	. . 3 ⊢ (𝐴 ≈ 𝐵 → Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = Scott {𝑥 ∣ 𝑥 ≈ 𝐵})
32	31, 5, 22	3eqtr4g 2820	. 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 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  Vcvv 3450  ∅c0 4278   class class class wbr 5102   ≈ cen 8973  Scott cscott 9931

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-en 8977  df-r1 9771  df-rank 9772  df-scott 9932

This theorem is used by:  kardeng  35881