Theorem catprslem

Description: Lemma for catprs . (Contributed by Zhi Wang, 18-Sep-2024)

	Ref	Expression
Hypotheses	catprs.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 𝐻 𝑦 ) ≠ ∅ ) )
	catprslem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	catprslem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	catprslem	⊢ ( 𝜑 → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		catprs.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 𝐻 𝑦 ) ≠ ∅ ) )
2		catprslem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
3		catprslem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
4		breq1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦 ) )
5		oveq1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 𝐻 𝑦 ) = ( 𝑧 𝐻 𝑦 ) )
6	5	neeq1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 𝐻 𝑦 ) ≠ ∅ ↔ ( 𝑧 𝐻 𝑦 ) ≠ ∅ ) )
7	4 6	bibi12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 𝐻 𝑦 ) ≠ ∅ ) ↔ ( 𝑧 ≤ 𝑦 ↔ ( 𝑧 𝐻 𝑦 ) ≠ ∅ ) ) )
8		breq2	⊢ ( 𝑦 = 𝑤 → ( 𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑤 ) )
9		oveq2	⊢ ( 𝑦 = 𝑤 → ( 𝑧 𝐻 𝑦 ) = ( 𝑧 𝐻 𝑤 ) )
10	9	neeq1d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑧 𝐻 𝑦 ) ≠ ∅ ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) )
11	8 10	bibi12d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑧 ≤ 𝑦 ↔ ( 𝑧 𝐻 𝑦 ) ≠ ∅ ) ↔ ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) ) )
12	7 11	cbvral2vw	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 𝐻 𝑦 ) ≠ ∅ ) ↔ ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) )
13	1 12	sylib	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) )
14		breq12	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) → ( 𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑌 ) )
15		oveq12	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) → ( 𝑧 𝐻 𝑤 ) = ( 𝑋 𝐻 𝑌 ) )
16	15	neeq1d	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) → ( ( 𝑧 𝐻 𝑤 ) ≠ ∅ ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) )
17	14 16	bibi12d	⊢ ( ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) → ( ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) ↔ ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) ) )
18	17	rspc2gv	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) ) )
19	2 3 18	syl2anc	⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑧 ≤ 𝑤 ↔ ( 𝑧 𝐻 𝑤 ) ≠ ∅ ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) ) )
20	13 19	mpd	⊢ ( 𝜑 → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 𝐻 𝑌 ) ≠ ∅ ) )