Theorem karden 9989

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 10665). 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 9987 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 5109	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	1	enref 9043	. . . . . 6 ⊢ 𝐴 ≈ 𝐴
4	1, 2, 3	ceqsexv2d 3503	. . . . 5 ⊢ ∃𝑥 𝑥 ≈ 𝐴
5		karden.c	. . . . . . 7 ⊢ 𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
6	5	neeq1i 3023	. . . . . 6 ⊢ (𝐶 ≠ ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ≠ ∅)
7		scott0b 9967	. . . . . . 7 ⊢ ({𝑥 ∣ 𝑥 ≈ 𝐴} = ∅ ↔ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = ∅)
8	7	necon3bii 3011	. . . . . 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 2853	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
15	14	pm4.71da 573	. . . . 5 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
16		breq1 5109	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
17	16	elscottab 9972	. . . . . . . 8 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐴} → 𝑦 ≈ 𝐴)
18	17, 5	eleq2s 2882	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ≈ 𝐴)
19	18	ensymd 9063	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ≈ 𝑦)
20		breq1 5109	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝐵 ↔ 𝑦 ≈ 𝐵))
21	20	elscottab 9972	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑥 ∣ 𝑥 ≈ 𝐵} → 𝑦 ≈ 𝐵)
22		karden.d	. . . . . . 7 ⊢ 𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}
23	21, 22	eleq2s 2882	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ≈ 𝐵)
24		entr 9064	. . . . . 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 1968	. . 3 ⊢ (𝐶 = 𝐷 → (∃𝑦 𝑦 ∈ 𝐶 → 𝐴 ≈ 𝐵))
28	13, 27	mpi 21	. 2 ⊢ (𝐶 = 𝐷 → 𝐴 ≈ 𝐵)
29		enen2 9168	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ≈ 𝐴 ↔ 𝑥 ≈ 𝐵))
30	29	abbidv 2829	. . . 4 ⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ 𝑥 ≈ 𝐴} = {𝑥 ∣ 𝑥 ≈ 𝐵})
31	30	scotteqd 9960	. . 3 ⊢ (𝐴 ≈ 𝐵 → Scott {𝑥 ∣ 𝑥 ≈ 𝐴} = Scott {𝑥 ∣ 𝑥 ≈ 𝐵})
32	31, 5, 22	3eqtr4g 2823	. 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 2147  {cab 2741   ≠ wne 2959  Vcvv 3454  ∅c0 4281   class class class wbr 5106   ≈ cen 9001  Scott cscott 9958

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-om 7905  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-en 9005  df-r1 9798  df-rank 9799  df-scott 9959

This theorem is used by:  kardeng  35938