Theorem kard0 35822

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 5261	. 2 ⊢ ∅ ∈ V
2		breq2 5107	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ ∅))
3	2	abbidv 2827	. . . . 5 ⊢ (𝑥 = ∅ → {𝑦 ∣ 𝑦 ≈ 𝑥} = {𝑦 ∣ 𝑦 ≈ ∅})
4	3	scotteqd 9930	. . . 4 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = Scott {𝑦 ∣ 𝑦 ≈ ∅})
5		en0 9045	. . . . . . . . . 10 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 = ∅)
6		velsn 4600	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
7	5, 6	bitr4i 281	. . . . . . . . 9 ⊢ (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅})
8	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑦 ≈ ∅ ↔ 𝑦 ∈ {∅}))
9	8	eqabcdv 2895	. . . . . . 7 ⊢ (⊤ → {𝑦 ∣ 𝑦 ≈ ∅} = {∅})
10	9	mptru 1577	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
11	10	scotteqi 35740	. . . . 5 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = Scott {∅}
12		scottsn 35750	. . . . 5 ⊢ Scott {∅} = {∅}
13	11, 12	eqtri 2784	. . . 4 ⊢ Scott {𝑦 ∣ 𝑦 ≈ ∅} = {∅}
14	4, 13	eqtrdi 2812	. . 3 ⊢ (𝑥 = ∅ → Scott {𝑦 ∣ 𝑦 ≈ 𝑥} = {∅})
15		df-kard 35818	. . 3 ⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
16		snex 5397	. . 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 2145  {cab 2739  Vcvv 3451  ∅c0 4279  {csn 4584   class class class wbr 5103  ‘cfv 6538   ≈ cen 8970  Scott cscott 9928  kardckard 35817

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  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-en 8974  df-scott 9929  df-kard 35818

This theorem is used by:  kard0b  35827