Theorem scottsn 35750

Description: Applying Scott's trick to a singleton leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)

Assertion

Ref	Expression
scottsn	⊢ Scott {𝐴} = {𝐴}

Proof of Theorem scottsn

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

Step	Hyp	Ref	Expression
1		df-scott 9929	. 2 ⊢ Scott {𝐴} = {𝑥 ∈ {𝐴} ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)}
2		velsn 4600	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 4600	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
4		eqtr3 2783	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
5	2, 3, 4	syl2anb 610	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
6		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
7	6	eqimssd 3987	. . . . 5 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) ⊆ (rank‘𝑦))
8	5, 7	syl 18	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (rank‘𝑥) ⊆ (rank‘𝑦))
9	8	ralrimiva 3155	. . 3 ⊢ (𝑥 ∈ {𝐴} → ∀𝑦 ∈ {𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))
10	9	rabeqc 3425	. 2 ⊢ {𝑥 ∈ {𝐴} ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝐴}
11	1, 10	eqtri 2784	1 ⊢ Scott {𝐴} = {𝐴}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  {csn 4584  ‘cfv 6538  rankcrnk 9767  Scott cscott 9928

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-ext 2733

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-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  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-iota 6494  df-fv 6546  df-scott 9929

This theorem is used by:  kard0  35822