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Theorem isline 38252
Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
isline.l ≀ = (leβ€˜πΎ)
isline.j ∨ = (joinβ€˜πΎ)
isline.a 𝐴 = (Atomsβ€˜πΎ)
isline.n 𝑁 = (Linesβ€˜πΎ)
Assertion
Ref Expression
isline (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
Distinct variable groups:   π‘ž,𝑝,π‘Ÿ,𝐴   𝐾,𝑝,π‘ž,π‘Ÿ   𝑋,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐷(π‘Ÿ,π‘ž,𝑝)   ∨ (π‘Ÿ,π‘ž,𝑝)   ≀ (π‘Ÿ,π‘ž,𝑝)   𝑁(π‘Ÿ,π‘ž,𝑝)   𝑋(𝑝)

Proof of Theorem isline
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 isline.l . . . 4 ≀ = (leβ€˜πΎ)
2 isline.j . . . 4 ∨ = (joinβ€˜πΎ)
3 isline.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 isline.n . . . 4 𝑁 = (Linesβ€˜πΎ)
51, 2, 3, 4lineset 38251 . . 3 (𝐾 ∈ 𝐷 β†’ 𝑁 = {π‘₯ ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})})
65eleq2d 2820 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {π‘₯ ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})}))
73fvexi 6860 . . . . . . . 8 𝐴 ∈ V
87rabex 5293 . . . . . . 7 {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} ∈ V
9 eleq1 2822 . . . . . . 7 (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} β†’ (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} ∈ V))
108, 9mpbiri 258 . . . . . 6 (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} β†’ 𝑋 ∈ V)
1110adantl 483 . . . . 5 ((π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) β†’ 𝑋 ∈ V)
1211a1i 11 . . . 4 ((π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) β†’ ((π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) β†’ 𝑋 ∈ V))
1312rexlimivv 3193 . . 3 (βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) β†’ 𝑋 ∈ V)
14 eqeq1 2737 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
1514anbi2d 630 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘ž β‰  π‘Ÿ ∧ π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) ↔ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
16152rexbidv 3210 . . 3 (π‘₯ = 𝑋 β†’ (βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}) ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
1713, 16elab3 3642 . 2 (𝑋 ∈ {π‘₯ ∣ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ π‘₯ = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})} ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
186, 17bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆƒwrex 3070  {crab 3406  Vcvv 3447   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  lecple 17148  joincjn 18208  Atomscatm 37775  Linesclines 38007
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-lines 38014
This theorem is referenced by:  islinei  38253  linepsubN  38265  isline2  38287
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