Theorem prslem 18323

Description: Lemma for prsref 18324 and prstr 18325. (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 18322	. . 3 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
4	3	simprbi 495	. 2 ⊢ (𝐾 ∈ Proset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
5		breq12 5158	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑥 = 𝑋) → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
6	5	anidms 565	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
7		breq1 5156	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
8	7	anbi1d 629	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧)))
9		breq1 5156	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
10	8, 9	imbi12d 343	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
11	6, 10	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
12		breq2 5157	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
13		breq1 5156	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
14	12, 13	anbi12d 630	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧)))
15	14	imbi1d 340	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
16	15	anbi2d 628	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
17		breq2 5157	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍))
18	17	anbi2d 628	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)))
19		breq2 5157	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
20	18, 19	imbi12d 343	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
21	20	anbi2d 628	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))))
22	11, 16, 21	rspc3v 3624	. 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 505	1 ⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3051  Vcvv 3462   class class class wbr 5153  ‘cfv 6554  Basecbs 17213  lecple 17273   Proset cproset 18318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-proset 18320

This theorem is referenced by:  prsref  18324  prstr  18325