Theorem catprslem 45907

Description: Lemma for catprs 45908. (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 5042	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
3		oveq1 7198	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
4	3	neeq1d 2991	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑦) ≠ ∅))
5	2, 4	bibi12d 349	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅)))
6		breq2 5043	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑤))
7		oveq2 7199	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
8	7	neeq1d 2991	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
9	6, 8	bibi12d 349	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)))
10	5, 9	cbvral2vw 3361	. . 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 5044	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑌))
15		oveq12 7200	. . . . . 6 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧𝐻𝑤) = (𝑋𝐻𝑌))
16	15	neeq1d 2991	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧𝐻𝑤) ≠ ∅ ↔ (𝑋𝐻𝑌) ≠ ∅))
17	14, 16	bibi12d 349	. . . 4 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) ↔ (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
18	17	rspc2gv 3536	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
19	12, 13, 18	syl2anc 587	. 2 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
20	11, 19	mpd 15	1 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051  ∅c0 4223   class class class wbr 5039  (class class class)co 7191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-ral 3056  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194

This theorem is referenced by:  catprs  45908