Theorem nfscott 9865

Description: Bound-variable hypothesis builder for the Scott operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypothesis

Ref	Expression
nfscott.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfscott	⊢ Ⅎ𝑥Scott 𝐴

Proof of Theorem nfscott

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

Step	Hyp	Ref	Expression
1		df-scott 9858	. 2 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
2		nfscott.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1941	. . . 4 ⊢ Ⅎ𝑥(rank‘𝑦) ⊆ (rank‘𝑧)
4	2, 3	nfralw 3318	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)
5	4, 2	nfrabw 3460	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
6	1, 5	nfcxfr 2929	1 ⊢ Ⅎ𝑥Scott 𝐴

Syntax hints:  Ⅎwnfc 2916  ∀wral 3085  {crab 3423   ⊆ wss 3913  ‘cfv 6537  rankcrnk 9735  Scott cscott 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rab 3424  df-scott 9858

This theorem is referenced by:  nfcoll  44892