Theorem nfscott 9857

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 9854	. 2 ⊢ Scott 𝐴 = {𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
2		nfscott.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1944	. . . 4 ⊢ Ⅎ𝑥(rank‘𝑦) ⊆ (rank‘𝑧)
4	2, 3	nfralw 3312	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)
5	4, 2	nfrabw 3452	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑦) ⊆ (rank‘𝑧)}
6	1, 5	nfcxfr 2923	1 ⊢ Ⅎ𝑥Scott 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2910  ∀wral 3079  {crab 3416   ⊆ wss 3905  ‘cfv 6536  rankcrnk 9731  Scott cscott 9853

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-scott 9854

This theorem is used by:  nfcoll  44994