Theorem catprslem 47708

Description: Lemma for catprs 47709. (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 5151	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
3		oveq1 7418	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
4	3	neeq1d 3000	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑦) ≠ ∅))
5	2, 4	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅)))
6		breq2 5152	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑤))
7		oveq2 7419	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
8	7	neeq1d 3000	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧𝐻𝑦) ≠ ∅ ↔ (𝑧𝐻𝑤) ≠ ∅))
9	6, 8	bibi12d 345	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑧 ≤ 𝑦 ↔ (𝑧𝐻𝑦) ≠ ∅) ↔ (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)))
10	5, 9	cbvral2vw 3238	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
11	1, 10	sylib 217	. 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 5153	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑌))
15		oveq12 7420	. . . . . 6 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → (𝑧𝐻𝑤) = (𝑋𝐻𝑌))
16	15	neeq1d 3000	. . . . 5 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧𝐻𝑤) ≠ ∅ ↔ (𝑋𝐻𝑌) ≠ ∅))
17	14, 16	bibi12d 345	. . . 4 ⊢ ((𝑧 = 𝑋 ∧ 𝑤 = 𝑌) → ((𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) ↔ (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
18	17	rspc2gv 3621	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)))
19	12, 13, 18	syl2anc 584	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∅c0 4322   class class class wbr 5148  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414

This theorem is referenced by:  catprs  47709