Theorem drsdir 18382

Description: Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

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

Assertion

Ref	Expression
drsdir	⊢ ((𝐾 ∈ Dirset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧))

Distinct variable groups:   𝑧,𝐾   𝑧,𝐵   𝑧, ≤   𝑧,𝑋   𝑧,𝑌

Proof of Theorem drsdir

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

Step	Hyp	Ref	Expression
1		isdrs.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		isdrs.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3	1, 2	isdrs 18381	. . . 4 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
4	3	simp3bi 1165	. . 3 ⊢ (𝐾 ∈ Dirset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))
5		breq1 5114	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
6	5	anbi1d 643	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
7	6	rexbidv 3191	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
8		breq1 5114	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
9	8	anbi2d 642	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
10	9	rexbidv 3191	. . . 4 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
11	7, 10	rspc2v 3594	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
12	4, 11	syl5com 32	. 2 ⊢ (𝐾 ∈ Dirset → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
13	12	3impib 1134	1 ⊢ ((𝐾 ∈ Dirset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  ∅c0 4286   class class class wbr 5111  ‘cfv 6540  Basecbs 17293  lecple 17341   Proset cproset 18372  Dirsetcdrs 18373

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  ax-nul 5271

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-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  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-drs 18375

This theorem is used by:  drsdirfi  18385