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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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