Theorem prslem 18319

Description: Lemma for prsref 18320 and prstr 18321. (Contributed by Mario Carneiro, 1-Feb-2015.)

Hypotheses

Ref	Expression
isprs.b	⊢ 𝐵 = (Base‘𝐾)
isprs.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
prslem	⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Proof of Theorem prslem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isprs.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		isprs.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3	1, 2	isprs 18318	. . 3 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
4	3	simprbi 501	. 2 ⊢ (𝐾 ∈ Proset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
5		breq12 5102	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
6	5	anidms 574	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
7		breq1 5100	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
8	7	anbi1d 640	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧)))
9		breq1 5100	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
10	8, 9	imbi12d 346	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
11	6, 10	anbi12d 641	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
12		breq2 5101	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
13		breq1 5100	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
14	12, 13	anbi12d 641	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧)))
15	14	imbi1d 343	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
16	15	anbi2d 639	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
17		breq2 5101	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍))
18	17	anbi2d 639	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)))
19		breq2 5101	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
20	18, 19	imbi12d 346	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
21	20	anbi2d 639	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))))
22	11, 16, 21	rspc3v 3596	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))))
23	4, 22	mpan9 514	1 ⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   class class class wbr 5097  ‘cfv 6515  Basecbs 17235  lecple 17283   Proset cproset 18314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6471  df-fv 6523  df-proset 18316

This theorem is referenced by:  prsref  18320  prstr  18321