Theorem nfscott 44208

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 44205	. 2 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
2		nfscott.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1913	. . . 4 ⊢ Ⅎ𝑥(rank‘𝑦) ⊆ (rank‘𝑧)
4	2, 3	nfralw 3317	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)
5	4, 2	nfrabw 3483	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
6	1, 5	nfcxfr 2906	1 ⊢ Ⅎ𝑥Scott 𝐴

Syntax hints:  Ⅎwnfc 2893  ∀wral 3067  {crab 3443   ⊆ wss 3976  ‘cfv 6573  rankcrnk 9832  Scott cscott 44204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-scott 44205

This theorem is referenced by:  nfcoll  44225