Theorem nfscott 44698

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 44695	. 2 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
2		nfscott.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1922	. . . 4 ⊢ Ⅎ𝑥(rank‘𝑦) ⊆ (rank‘𝑧)
4	2, 3	nfralw 3288	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)
5	4, 2	nfrabw 3430	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
6	1, 5	nfcxfr 2901	1 ⊢ Ⅎ𝑥Scott 𝐴

Syntax hints:  Ⅎwnfc 2888  ∀wral 3055  {crab 3393   ⊆ wss 3885  ‘cfv 6489  rankcrnk 9682  Scott cscott 44694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rab 3394  df-scott 44695

This theorem is referenced by:  nfcoll  44715