Theorem drsdir 18374

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 18373	. . . 4 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
4	3	simp3bi 1147	. . 3 ⊢ (𝐾 ∈ Dirset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧))
5		breq1 5169	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
6	5	anbi1d 630	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
7	6	rexbidv 3185	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧)))
8		breq1 5169	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
9	8	anbi2d 629	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
10	9	rexbidv 3185	. . . 4 ⊢ (𝑦 = 𝑌 → (∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ↔ ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
11	7, 10	rspc2v 3646	. . 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 31	. 2 ⊢ (𝐾 ∈ Dirset → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
13	12	3impib 1116	1 ⊢ ((𝐾 ∈ Dirset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∅c0 4352   class class class wbr 5166  ‘cfv 6575  Basecbs 17260  lecple 17320   Proset cproset 18365  Dirsetcdrs 18366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-drs 18368

This theorem is referenced by:  drsdirfi  18377