Theorem catprslem 49847

Description: Lemma for catprs 49848. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

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

Assertion

Ref	Expression
catprslem	⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))

Distinct variable groups:   𝑥, ≤ ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem catprslem

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catprs.1	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
2		breq1 5114	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
3		oveq1 7426	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
4	3	neeq1d 3019	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑦) ≠ ∅))
5	2, 4	bibi12d 348	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅)))
6		breq2 5115	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑤))
7		oveq2 7427	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
8	7	neeq1d 3019	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
9	6, 8	bibi12d 348	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)))
10	5, 9	cbvral2vw 3249	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
11	1, 10	sylib 221	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
12		catprslem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13		catprslem.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		breq12 5116	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑌))
15		oveq12 7428	. . . . . 6 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧𝐻𝑤) = (𝑋𝐻𝑌))
16	15	neeq1d 3019	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧𝐻𝑤) ≠ ∅ ↔ (𝑋𝐻𝑌) ≠ ∅))
17	14, 16	bibi12d 348	. . . 4 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) ↔ (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
18	17	rspc2gv 3593	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
19	12, 13, 18	syl2anc 596	. 2 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
20	11, 19	mpd 16	1 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∅c0 4286   class class class wbr 5111  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  catprs  49848