Theorem scottss 44670

Description: Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Assertion

Ref	Expression
scottss	⊢ Scott 𝐴 ⊆ 𝐴

Proof of Theorem scottss

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

Step	Hyp	Ref	Expression
1		df-scott 44663	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
2	1	ssrab3 4022	1 ⊢ Scott 𝐴 ⊆ 𝐴

Syntax hints:  ∀wral 3051   ⊆ wss 3889  ‘cfv 6498  rankcrnk 9687  Scott cscott 44662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-ss 3906  df-scott 44663

This theorem is referenced by:  elscottab  44671