Theorem kard0 35893

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 5263	. 2 ⊢ ∅ ∈ V
2		breq2 5109	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ ∅))
3	2	abbidv 2829	. . . . 5 ⊢ (𝑥 = ∅ → {𝑦 ∣ 𝑦 ≈ 𝑥} = {𝑦 ∣ 𝑦 ≈ ∅})
4	3	scotteqd 9940	. . . 4 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = Scott {𝑦 ∣ 𝑦 ≈ ∅})
5		en0 9056	. . . . . . . . . 10 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 = ∅)
6		velsn 4599	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
7	5, 6	bitr4i 281	. . . . . . . . 9 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅})
8	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅}))
9	8	eqabcdv 2897	. . . . . . 7 ⊢ (⊤ → {𝑦 ∣ 𝑦 ≈ ∅} = {∅})
10	9	mptru 1576	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
11	10	scotteqi 35803	. . . . 5 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = Scott {∅}
12		scottsn 35813	. . . . 5 ⊢ Scott {∅} = {∅}
13	11, 12	eqtri 2786	. . . 4 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
14	4, 13	eqtrdi 2814	. . 3 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = {∅})
15		df-kard 35889	. . 3 ⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
16		snex 5400	. . 3 ⊢ {∅} ∈ V
17	14, 15, 16	fvmpt 7001	. 2 ⊢ (∅ ∈ V → (kard‘∅) = {∅})
18	1, 17	ax-mp 5	1 ⊢ (kard‘∅) = {∅}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ⊤wtru 1570   ∈ wcel 2146  {cab 2741  Vcvv 3453  ∅c0 4280  {csn 4587   class class class wbr 5105  ‘cfv 6546   ≈ cen 8981  Scott cscott 9938  kardckard 35888

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-en 8985  df-scott 9939  df-kard 35889

This theorem is used by:  kard0b  35898