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Theorem drsdir 18255
Description: Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
isdrs.b 𝐡 = (Baseβ€˜πΎ)
isdrs.l ≀ = (leβ€˜πΎ)
Assertion
Ref Expression
drsdir ((𝐾 ∈ Dirset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧))
Distinct variable groups:   𝑧,𝐾   𝑧,𝐡   𝑧, ≀   𝑧,𝑋   𝑧,π‘Œ
Proof of Theorem drsdir
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdrs.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 isdrs.l . . . . 5 ≀ = (leβ€˜πΎ)
31, 2isdrs 18254 . . . 4 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐡 β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (π‘₯ ≀ 𝑧 ∧ 𝑦 ≀ 𝑧)))
43simp3bi 1148 . . 3 (𝐾 ∈ Dirset β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (π‘₯ ≀ 𝑧 ∧ 𝑦 ≀ 𝑧))
5 breq1 5152 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑧 ↔ 𝑋 ≀ 𝑧))
65anbi1d 631 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑧 ∧ 𝑦 ≀ 𝑧) ↔ (𝑋 ≀ 𝑧 ∧ 𝑦 ≀ 𝑧)))
76rexbidv 3179 . . . 4 (π‘₯ = 𝑋 β†’ (βˆƒπ‘§ ∈ 𝐡 (π‘₯ ≀ 𝑧 ∧ 𝑦 ≀ 𝑧) ↔ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ 𝑦 ≀ 𝑧)))
8 breq1 5152 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑦 ≀ 𝑧 ↔ π‘Œ ≀ 𝑧))
98anbi2d 630 . . . . 5 (𝑦 = π‘Œ β†’ ((𝑋 ≀ 𝑧 ∧ 𝑦 ≀ 𝑧) ↔ (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
109rexbidv 3179 . . . 4 (𝑦 = π‘Œ β†’ (βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ 𝑦 ≀ 𝑧) ↔ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
117, 10rspc2v 3623 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (π‘₯ ≀ 𝑧 ∧ 𝑦 ≀ 𝑧) β†’ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
124, 11syl5com 31 . 2 (𝐾 ∈ Dirset β†’ ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
13123impib 1117 1 ((𝐾 ∈ Dirset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐡 (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204   Proset cproset 18246  Dirsetcdrs 18247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-drs 18249
This theorem is referenced by:  drsdirfi  18258
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