Theorem tghilberti1 28709

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 28702	. 2 ⊢ (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
9	1, 2, 3, 4, 5, 6, 7	tglinerflx1 28705	. 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))
10	1, 2, 3, 4, 5, 6, 7	tglinerflx2 28706	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))
11		eleq2 2825	. . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑃𝐿𝑄)))
12		eleq2 2825	. . . 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 3576	. 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 2932  ∃wrex 3060  ran crn 5625  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  TarskiGcstrkg 28499  Itvcitv 28505  LineGclng 28506

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-trkgc 28520  df-trkgb 28521  df-trkgcb 28522  df-trkg 28525

This theorem is referenced by:  tglinethrueu  28711