Theorem tghilberti1 28658

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	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tghilberti1	⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥,𝐼   𝑥,𝐿   𝑥,𝑃   𝑥,𝑄   𝜑,𝑥

Proof of Theorem tghilberti1

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineelsb2.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglineelsb2.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
6		tglineelsb2.2	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
7		tglineelsb2.4	. . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
8	1, 2, 3, 4, 5, 6, 7	tgelrnln 28651	. 2 ⊢ (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
9	1, 2, 3, 4, 5, 6, 7	tglinerflx1 28654	. 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))
10	1, 2, 3, 4, 5, 6, 7	tglinerflx2 28655	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))
11		eleq2 2823	. . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑃𝐿𝑄)))
12		eleq2 2823	. . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ (𝑃𝐿𝑄)))
13	11, 12	anbi12d 632	. . 3 ⊢ (𝑥 = (𝑃𝐿𝑄) → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
14	13	rspcev 3574	. 2 ⊢ (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
15	8, 9, 10, 14	syl12anc 836	1 ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  ran crn 5623  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  TarskiGcstrkg 28448  Itvcitv 28454  LineGclng 28455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-trkgc 28469  df-trkgb 28470  df-trkgcb 28471  df-trkg 28474

This theorem is referenced by:  tglinethrueu  28660