Theorem scottsn 35816

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 9942	. 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 2785	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐴) → 𝑥 = 𝑦)
5	2, 3, 4	syl2anb 610	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
6		fveq2 6896	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
7	6	eqimssd 3969	. . . . 5 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) ⊆ (rank‘𝑦))
8	5, 7	syl 18	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (rank‘𝑥) ⊆ (rank‘𝑦))
9	8	ralrimiva 3158	. . 3 ⊢ (𝑥 ∈ {𝐴} → ∀𝑦 ∈ {𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))
10	9	rabeqc 3428	. 2 ⊢ {𝑥 ∈ {𝐴} ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝐴}
11	1, 10	eqtri 2786	1 ⊢ Scott {𝐴} = {𝐴}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3080  {crab 3416   ⊆ wss 3900  {csn 4588  ‘cfv 6547  rankcrnk 9780  Scott cscott 9941

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-ext 2735

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-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-scott 9942

This theorem is used by:  kard0  35896