Theorem hilbert1.2 36188

Description: There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)

Assertion

Ref	Expression
hilbert1.2	⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Distinct variable groups:   𝑥,𝑃   𝑥,𝑄

Proof of Theorem hilbert1.2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 656	. . . . 5 ⊢ (((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) ↔ ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))))
2		simprl 770	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 ∈ LinesEE)
3		simprr 772	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
4		simpl 482	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑃 ≠ 𝑄)
5		linethru 36186	. . . . . . . . 9 ⊢ ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ 𝑃 ≠ 𝑄) → 𝑥 = (𝑃Line𝑄))
6	2, 3, 4, 5	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 = (𝑃Line𝑄))
7	6	ex 412	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) → 𝑥 = (𝑃Line𝑄)))
8		simprl 770	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ LinesEE)
9		simprr 772	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))
10		simpl 482	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
11		linethru 36186	. . . . . . . . 9 ⊢ ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦) ∧ 𝑃 ≠ 𝑄) → 𝑦 = (𝑃Line𝑄))
12	8, 9, 10, 11	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃Line𝑄))
13	12	ex 412	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑦 = (𝑃Line𝑄)))
14	7, 13	anim12d 609	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑥 = (𝑃Line𝑄) ∧ 𝑦 = (𝑃Line𝑄))))
15		eqtr3 2753	. . . . . 6 ⊢ ((𝑥 = (𝑃Line𝑄) ∧ 𝑦 = (𝑃Line𝑄)) → 𝑥 = 𝑦)
16	14, 15	syl6 35	. . . . 5 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
17	1, 16	biimtrid 242	. . . 4 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
18	17	expd 415	. . 3 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦)))
19	18	ralrimivv 3173	. 2 ⊢ (𝑃 ≠ 𝑄 → ∀𝑥 ∈ LinesEE ∀𝑦 ∈ LinesEE (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
20		eleq2w 2815	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
21		eleq2w 2815	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
22	20, 21	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
23	22	rmo4 3689	. 2 ⊢ (∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ ∀𝑥 ∈ LinesEE ∀𝑦 ∈ LinesEE (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
24	19, 23	sylibr 234	1 ⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃*wrmo 3345  (class class class)co 7346  Linecline2 36167  LinesEEclines2 36169

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-ec 8624  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-oi 9396  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-n0 12379  df-z 12466  df-uz 12730  df-rp 12888  df-ico 13248  df-icc 13249  df-fz 13405  df-fzo 13552  df-seq 13906  df-exp 13966  df-hash 14235  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-clim 15392  df-sum 15591  df-ee 28867  df-btwn 28868  df-cgr 28869  df-ofs 36016  df-colinear 36072  df-ifs 36073  df-cgr3 36074  df-fs 36075  df-line2 36170  df-lines2 36172

This theorem is referenced by:  linethrueu  36189