Theorem tgisline 28653

Description: The property of being a proper line, generated by two distinct points. (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)
tgisline.1	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Assertion

Ref	Expression
tgisline	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐿(𝑥,𝑦)

Proof of Theorem tgisline

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
3		tglineelsb2.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
4		tglineelsb2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝐺 ∈ TarskiG)
6		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥 ∈ 𝐵)
7		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ (𝐵 ∖ {𝑥}))
8	7	eldifad 3988	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ∈ 𝐵)
9		eldifsn 4811	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 𝑥))
10	7, 9	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 𝑥))
11	10	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑦 ≠ 𝑥)
12	11	necomd 3002	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → 𝑥 ≠ 𝑦)
13	1, 2, 3, 5, 6, 8, 12	tglngval 28577	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → (𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
14	13, 12	jca 511	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ {𝑥}))) → ((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦))
15	14	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐵 ∖ {𝑥})((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦))
16		tgisline.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
17	1, 2, 3	tglng 28572	. . . . . . 7 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
18	4, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿 = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
19	18	rneqd 5963	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
20	16, 19	eleqtrd 2846	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
21		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
22	21	elrnmpog 7585	. . . . 5 ⊢ (𝐴 ∈ ran 𝐿 → (𝐴 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
23	16, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
24	20, 23	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ (𝐵 ∖ {𝑥})𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
25	15, 24	r19.29d2r 3146	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
26		difss 4159	. . . 4 ⊢ (𝐵 ∖ {𝑥}) ⊆ 𝐵
27		simpr 484	. . . . . . 7 ⊢ ((((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
28		simpll 766	. . . . . . 7 ⊢ ((((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
29	27, 28	eqtr4d 2783	. . . . . 6 ⊢ ((((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝐴 = (𝑥𝐿𝑦))
30		simplr 768	. . . . . 6 ⊢ ((((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑥 ≠ 𝑦)
31	29, 30	jca 511	. . . . 5 ⊢ ((((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
32	31	reximi 3090	. . . 4 ⊢ (∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑦 ∈ (𝐵 ∖ {𝑥})(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
33		ssrexv 4078	. . . 4 ⊢ ((𝐵 ∖ {𝑥}) ⊆ 𝐵 → (∃𝑦 ∈ (𝐵 ∖ {𝑥})(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦) → ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)))
34	26, 32, 33	mpsyl 68	. . 3 ⊢ (∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
35	34	reximi 3090	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ (𝐵 ∖ {𝑥})(((𝑥𝐿𝑦) = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∧ 𝑥 ≠ 𝑦) ∧ 𝐴 = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
36	25, 35	syl 17	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Basecbs 17258  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-trkg 28479

This theorem is referenced by:  tglnne  28654  tglndim0  28655  tglinethru  28662  tglnne0  28666  tglnpt2  28667  footexALT  28744  footex  28747  opptgdim2  28771