Theorem hilbert1.2 36469

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 666	. . . . 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 780	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 ∈ LinesEE)
3		simprr 782	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
4		simpl 486	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑃 ≠ 𝑄)
5		linethru 36467	. . . . . . . . 9 ⊢ ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ 𝑃 ≠ 𝑄) → 𝑥 = (𝑃Line𝑄))
6	2, 3, 4, 5	syl3anc 1389	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))) → 𝑥 = (𝑃Line𝑄))
7	6	ex 416	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) → 𝑥 = (𝑃Line𝑄)))
8		simprl 780	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ LinesEE)
9		simprr 782	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))
10		simpl 486	. . . . . . . . 9 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
11		linethru 36467	. . . . . . . . 9 ⊢ ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦) ∧ 𝑃 ≠ 𝑄) → 𝑦 = (𝑃Line𝑄))
12	8, 9, 10, 11	syl3anc 1389	. . . . . . . 8 ⊢ ((𝑃 ≠ 𝑄 ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃Line𝑄))
13	12	ex 416	. . . . . . 7 ⊢ (𝑃 ≠ 𝑄 → ((𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑦 = (𝑃Line𝑄)))
14	7, 13	anim12d 618	. . . . . 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 35	. . . . 5 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) ∧ (𝑦 ∈ LinesEE ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
17	1, 16	biimtrid 244	. . . 4 ⊢ (𝑃 ≠ 𝑄 → (((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦))
18	17	expd 419	. . 3 ⊢ (𝑃 ≠ 𝑄 → ((𝑥 ∈ LinesEE ∧ 𝑦 ∈ LinesEE) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦)))
19	18	ralrimivv 3202	. 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 641	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
23	22	rmo4 3692	. 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 236	1 ⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃*wrmo 3365  (class class class)co 7392  Linecline2 36448  LinesEEclines2 36450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-ec 8675  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-ico 13352  df-icc 13353  df-fz 13510  df-fzo 13657  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-sum 15697  df-ee 29037  df-btwn 29038  df-cgr 29039  df-ofs 36297  df-colinear 36353  df-ifs 36354  df-cgr3 36355  df-fs 36356  df-line2 36451  df-lines2 36453

This theorem is referenced by:  linethrueu  36470