Theorem drsdir 18393

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 18392	. . . 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 5106	. . . . . 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 5106	. . . . . 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 3587	. . 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 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4279   class class class wbr 5103  ‘cfv 6533  Basecbs 17304  lecple 17352   Proset cproset 18383  Dirsetcdrs 18384

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 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-drs 18386

This theorem is used by:  drsdirfi  18396