Theorem hilbert1.2 36890

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 669	. . . . 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 783	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 ∈ LinesEE)
3		simprr 785	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
4		simpl 488	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑃 ≠ 𝑄)
5		linethru 36888	. . . . . . . . 9 ⊢ ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ 𝑃 ≠ 𝑄) → 𝑥 = (𝑃Line𝑄))
6	2, 3, 4, 5	syl3anc 1398	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 = (𝑃Line𝑄))
7	6	ex 418	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) → 𝑥 = (𝑃Line𝑄)))
8		simprl 783	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ LinesEE)
9		simprr 785	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))
10		simpl 488	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
11		linethru 36888	. . . . . . . . 9 ⊢ ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦) ∧ 𝑃 ≠ 𝑄) → 𝑦 = (𝑃Line𝑄))
12	8, 9, 10, 11	syl3anc 1398	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃Line𝑄))
13	12	ex 418	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑦 = (𝑃Line𝑄)))
14	7, 13	anim12d 621	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑥 = (𝑃Line𝑄) ∧ 𝑦 = (𝑃Line𝑄))))
15		eqtr3 2783	. . . . . 6 ⊢ ((𝑥 = (𝑃Line𝑄) ∧ 𝑦 = (𝑃Line𝑄)) → 𝑥 = 𝑦)
16	14, 15	syl6 36	. . . . 5 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
17	1, 16	biimtrid 245	. . . 4 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
18	17	expd 421	. . 3 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦)))
19	18	ralrimivv 3204	. 2 ⊢ (𝑃 ≠ 𝑄 → ∀𝑥 ∈ LinesEE ∀𝑦 ∈ LinesEE (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
20		eleq2w 2845	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
21		eleq2w 2845	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
22	20, 21	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
23	22	rmo4 3688	. 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 237	1 ⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃*wrmo 3365  (class class class)co 7412  Linecline2 36869  LinesEEclines2 36871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-ec 8703  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-ico 13463  df-icc 13464  df-fz 13621  df-fzo 13769  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-sum 15834  df-ee 29450  df-btwn 29451  df-cgr 29452  df-ofs 36718  df-colinear 36774  df-ifs 36775  df-cgr3 36776  df-fs 36777  df-line2 36872  df-lines2 36874

This theorem is used by:  linethrueu  36891