Theorem kard0 35500

Description: The kard cardinality of the empty set is the singleton of the empty set. (Contributed by BTernaryTau, 3-Jul-2026.)

Assertion

Ref	Expression
kard0	⊢ (kard‘∅) = {∅}

Proof of Theorem kard0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5272	. 2 ⊢ ∅ ∈ V
2		breq2 5117	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ ∅))
3	2	abbidv 2835	. . . . 5 ⊢ (𝑥 = ∅ → {𝑦 ∣ 𝑦 ≈ 𝑥} = {𝑦 ∣ 𝑦 ≈ ∅})
4	3	scotteqd 9863	. . . 4 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = Scott {𝑦 ∣ 𝑦 ≈ ∅})
5		en0 9015	. . . . . . . . . 10 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 = ∅)
6		velsn 4610	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
7	5, 6	bitr4i 281	. . . . . . . . 9 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅})
8	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅}))
9	8	eqabcdv 2903	. . . . . . 7 ⊢ (⊤ → {𝑦 ∣ 𝑦 ≈ ∅} = {∅})
10	9	mptru 1574	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
11	10	scotteqi 35450	. . . . 5 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = Scott {∅}
12		scottsn 35454	. . . . 5 ⊢ Scott {∅} = {∅}
13	11, 12	eqtri 2792	. . . 4 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
14	4, 13	eqtrdi 2820	. . 3 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = {∅})
15		df-kard 35496	. . 3 ⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
16		snex 5411	. . 3 ⊢ {∅} ∈ V
17	14, 15, 16	fvmpt 6990	. 2 ⊢ (∅ ∈ V → (kard‘∅) = {∅})
18	1, 17	ax-mp 5	1 ⊢ (kard‘∅) = {∅}

Syntax hints:   ↔ wb 209   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  {cab 2747  Vcvv 3463  ∅c0 4294  {csn 4594   class class class wbr 5113  ‘cfv 6537   ≈ cen 8940  Scott cscott 9857  kardckard 35495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  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-en 8944  df-scott 9858  df-kard 35496

This theorem is referenced by:  kard0b  35505