Theorem nfscott 41857

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 41854	. 2 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
2		nfscott.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1917	. . . 4 ⊢ Ⅎ𝑥(rank‘𝑦) ⊆ (rank‘𝑧)
4	2, 3	nfralw 3151	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)
5	4, 2	nfrabw 3318	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
6	1, 5	nfcxfr 2905	1 ⊢ Ⅎ𝑥Scott 𝐴

Syntax hints:  Ⅎwnfc 2887  ∀wral 3064  {crab 3068   ⊆ wss 3887  ‘cfv 6433  rankcrnk 9521  Scott cscott 41853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-scott 41854

This theorem is referenced by:  nfcoll  41874