Theorem kard0 35626

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 5115	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ ∅))
3	2	abbidv 2831	. . . . 5 ⊢ (𝑥 = ∅ → {𝑦 ∣ 𝑦 ≈ 𝑥} = {𝑦 ∣ 𝑦 ≈ ∅})
4	3	scotteqd 9866	. . . 4 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = Scott {𝑦 ∣ 𝑦 ≈ ∅})
5		en0 9021	. . . . . . . . . 10 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 = ∅)
6		velsn 4607	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
7	5, 6	bitr4i 281	. . . . . . . . 9 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅})
8	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅}))
9	8	eqabcdv 2899	. . . . . . 7 ⊢ (⊤ → {𝑦 ∣ 𝑦 ≈ ∅} = {∅})
10	9	mptru 1577	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
11	10	scotteqi 35569	. . . . 5 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = Scott {∅}
12		scottsn 35579	. . . . 5 ⊢ Scott {∅} = {∅}
13	11, 12	eqtri 2788	. . . 4 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
14	4, 13	eqtrdi 2816	. . 3 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = {∅})
15		df-kard 35622	. . 3 ⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
16		snex 5412	. . 3 ⊢ {∅} ∈ V
17	14, 15, 16	fvmpt 6993	. 2 ⊢ (∅ ∈ V → (kard‘∅) = {∅})
18	1, 17	ax-mp 5	1 ⊢ (kard‘∅) = {∅}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  {cab 2743  Vcvv 3457  ∅c0 4286  {csn 4591   class class class wbr 5111  ‘cfv 6540   ≈ cen 8946  Scott cscott 9864  kardckard 35621

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 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-en 8950  df-scott 9865  df-kard 35622

This theorem is used by:  kard0b  35631