Theorem hilbert1.2 36337

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 657	. . . . 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 771	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 ∈ LinesEE)
3		simprr 773	. . . . . . . . 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 36335	. . . . . . . . 9 ⊢ ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ 𝑃 ≠ 𝑄) → 𝑥 = (𝑃Line𝑄))
6	2, 3, 4, 5	syl3anc 1374	. . . . . . . 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 771	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ LinesEE)
9		simprr 773	. . . . . . . . 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 36335	. . . . . . . . 9 ⊢ ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦) ∧ 𝑃 ≠ 𝑄) → 𝑦 = (𝑃Line𝑄))
12	8, 9, 10, 11	syl3anc 1374	. . . . . . . 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 610	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑥 = (𝑃Line𝑄) ∧ 𝑦 = (𝑃Line𝑄))))
15		eqtr3 2758	. . . . . 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 3178	. 2 ⊢ (𝑃 ≠ 𝑄 → ∀𝑥 ∈ LinesEE ∀𝑦 ∈ LinesEE (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
20		eleq2w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
21		eleq2w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
22	20, 21	anbi12d 633	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
23	22	rmo4 3676	. 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 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃*wrmo 3341  (class class class)co 7367  Linecline2 36316  LinesEEclines2 36318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-ec 8645  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649  df-ee 28959  df-btwn 28960  df-cgr 28961  df-ofs 36165  df-colinear 36221  df-ifs 36222  df-cgr3 36223  df-fs 36224  df-line2 36319  df-lines2 36321

This theorem is referenced by:  linethrueu  36338