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Theorem drsdir 16982
 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 ∧ 𝑋𝐵𝑌𝐵) → ∃𝑧𝐵 (𝑋 𝑧𝑌 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.
StepHypRef Expression
1 isdrs.b . . . . 5 𝐵 = (Base‘𝐾)
2 isdrs.l . . . . 5 = (le‘𝐾)
31, 2isdrs 16981 . . . 4 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑧𝑦 𝑧)))
43simp3bi 1098 . . 3 (𝐾 ∈ Dirset → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑧𝑦 𝑧))
5 breq1 4688 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑧𝑋 𝑧))
65anbi1d 741 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑧𝑦 𝑧) ↔ (𝑋 𝑧𝑦 𝑧)))
76rexbidv 3081 . . . 4 (𝑥 = 𝑋 → (∃𝑧𝐵 (𝑥 𝑧𝑦 𝑧) ↔ ∃𝑧𝐵 (𝑋 𝑧𝑦 𝑧)))
8 breq1 4688 . . . . . 6 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
98anbi2d 740 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑧𝑦 𝑧) ↔ (𝑋 𝑧𝑌 𝑧)))
109rexbidv 3081 . . . 4 (𝑦 = 𝑌 → (∃𝑧𝐵 (𝑋 𝑧𝑦 𝑧) ↔ ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧)))
117, 10rspc2v 3353 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑧𝑦 𝑧) → ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧)))
124, 11syl5com 31 . 2 (𝐾 ∈ Dirset → ((𝑋𝐵𝑌𝐵) → ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧)))
13123impib 1281 1 ((𝐾 ∈ Dirset ∧ 𝑋𝐵𝑌𝐵) → ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  ∅c0 3948   class class class wbr 4685  ‘cfv 5926  Basecbs 15904  lecple 15995   Preset cpreset 16973  Dirsetcdrs 16974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-drs 16976 This theorem is referenced by:  drsdirfi  16985
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