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Theorem tghilberti1 27885
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐡 = (Baseβ€˜πΊ)
tglineelsb2.i 𝐼 = (Itvβ€˜πΊ)
tglineelsb2.l 𝐿 = (LineGβ€˜πΊ)
tglineelsb2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglineelsb2.1 (πœ‘ β†’ 𝑃 ∈ 𝐡)
tglineelsb2.2 (πœ‘ β†’ 𝑄 ∈ 𝐡)
tglineelsb2.4 (πœ‘ β†’ 𝑃 β‰  𝑄)
Assertion
Ref Expression
tghilberti1 (πœ‘ β†’ βˆƒπ‘₯ ∈ ran 𝐿(𝑃 ∈ π‘₯ ∧ 𝑄 ∈ π‘₯))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐺   π‘₯,𝐼   π‘₯,𝐿   π‘₯,𝑃   π‘₯,𝑄   πœ‘,π‘₯

Proof of Theorem tghilberti1
StepHypRef Expression
1 tglineelsb2.p . . 3 𝐡 = (Baseβ€˜πΊ)
2 tglineelsb2.i . . 3 𝐼 = (Itvβ€˜πΊ)
3 tglineelsb2.l . . 3 𝐿 = (LineGβ€˜πΊ)
4 tglineelsb2.g . . 3 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 tglineelsb2.1 . . 3 (πœ‘ β†’ 𝑃 ∈ 𝐡)
6 tglineelsb2.2 . . 3 (πœ‘ β†’ 𝑄 ∈ 𝐡)
7 tglineelsb2.4 . . 3 (πœ‘ β†’ 𝑃 β‰  𝑄)
81, 2, 3, 4, 5, 6, 7tgelrnln 27878 . 2 (πœ‘ β†’ (𝑃𝐿𝑄) ∈ ran 𝐿)
91, 2, 3, 4, 5, 6, 7tglinerflx1 27881 . 2 (πœ‘ β†’ 𝑃 ∈ (𝑃𝐿𝑄))
101, 2, 3, 4, 5, 6, 7tglinerflx2 27882 . 2 (πœ‘ β†’ 𝑄 ∈ (𝑃𝐿𝑄))
11 eleq2 2822 . . . 4 (π‘₯ = (𝑃𝐿𝑄) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃𝐿𝑄)))
12 eleq2 2822 . . . 4 (π‘₯ = (𝑃𝐿𝑄) β†’ (𝑄 ∈ π‘₯ ↔ 𝑄 ∈ (𝑃𝐿𝑄)))
1311, 12anbi12d 631 . . 3 (π‘₯ = (𝑃𝐿𝑄) β†’ ((𝑃 ∈ π‘₯ ∧ 𝑄 ∈ π‘₯) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
1413rspcev 3612 . 2 (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) β†’ βˆƒπ‘₯ ∈ ran 𝐿(𝑃 ∈ π‘₯ ∧ 𝑄 ∈ π‘₯))
158, 9, 10, 14syl12anc 835 1 (πœ‘ β†’ βˆƒπ‘₯ ∈ ran 𝐿(𝑃 ∈ π‘₯ ∧ 𝑄 ∈ π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  TarskiGcstrkg 27675  Itvcitv 27681  LineGclng 27682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-trkgc 27696  df-trkgb 27697  df-trkgcb 27698  df-trkg 27701
This theorem is referenced by:  tglinethrueu  27887
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