Theorem scott0f

Description: A version of scott0 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018)

	Ref	Expression
Hypotheses	scott0f.1	⊢ Ⅎ 𝑦 𝐴
	scott0f.2	⊢ Ⅎ 𝑥 𝐴
Assertion	scott0f	⊢ ( 𝐴 = ∅ ↔ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = ∅ )

Proof

Step	Hyp	Ref	Expression
1		scott0f.1	⊢ Ⅎ 𝑦 𝐴
2		scott0f.2	⊢ Ⅎ 𝑥 𝐴
3		scott0	⊢ ( 𝐴 = ∅ ↔ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) } = ∅ )
4		nfcv	⊢ Ⅎ 𝑧 𝐴
5		nfv	⊢ Ⅎ 𝑧 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 )
6		nfv	⊢ Ⅎ 𝑦 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 )
7		fveq2	⊢ ( 𝑦 = 𝑧 → ( rank ‘ 𝑦 ) = ( rank ‘ 𝑧 ) )
8	7	sseq2d	⊢ ( 𝑦 = 𝑧 → ( ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) ↔ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) ) )
9	1 4 5 6 8	cbvralfw	⊢ ( ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) )
10	9	rabbii	⊢ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) }
11		nfcv	⊢ Ⅎ 𝑤 𝐴
12		nfv	⊢ Ⅎ 𝑥 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 )
13	2 12	nfralw	⊢ Ⅎ 𝑥 ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 )
14		nfv	⊢ Ⅎ 𝑤 ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 )
15		fveq2	⊢ ( 𝑤 = 𝑥 → ( rank ‘ 𝑤 ) = ( rank ‘ 𝑥 ) )
16	15	sseq1d	⊢ ( 𝑤 = 𝑥 → ( ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) ↔ ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) ) )
17	16	ralbidv	⊢ ( 𝑤 = 𝑥 → ( ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) ) )
18	11 2 13 14 17	cbvrabw	⊢ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) } = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑧 ) }
19	10 18	eqtr4i	⊢ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) }
20	19	eqeq1i	⊢ ( { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = ∅ ↔ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑧 ∈ 𝐴 ( rank ‘ 𝑤 ) ⊆ ( rank ‘ 𝑧 ) } = ∅ )
21	3 20	bitr4i	⊢ ( 𝐴 = ∅ ↔ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = ∅ )