Theorem drsdir 18476

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 18475	. . . 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 5105	. . . . . 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 3186	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
8		breq1 5105	. . . . . 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 3186	. . . 4 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
11	7, 10	rspc2v 3586	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278   class class class wbr 5102  ‘cfv 6538  Basecbs 17387  lecple 17435   Proset cproset 18466  Dirsetcdrs 18467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-drs 18469

This theorem is used by:  drsdirfi  18479