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Theorem hilbert1.1 31956
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
hilbert1.1 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
Distinct variable groups:   𝑥,𝑃   𝑥,𝑄
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem hilbert1.1
Dummy variables 𝑛 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1059 . . . . 5 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) → 𝑃 ∈ (𝔼‘𝑁))
2 simp2 1060 . . . . 5 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) → 𝑄 ∈ (𝔼‘𝑁))
3 simp3 1061 . . . . 5 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) → 𝑃𝑄)
4 eqidd 2622 . . . . 5 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) → (𝑃Line𝑄) = (𝑃Line𝑄))
5 neeq1 2852 . . . . . . 7 (𝑝 = 𝑃 → (𝑝𝑞𝑃𝑞))
6 oveq1 6622 . . . . . . . 8 (𝑝 = 𝑃 → (𝑝Line𝑞) = (𝑃Line𝑞))
76eqeq2d 2631 . . . . . . 7 (𝑝 = 𝑃 → ((𝑃Line𝑄) = (𝑝Line𝑞) ↔ (𝑃Line𝑄) = (𝑃Line𝑞)))
85, 7anbi12d 746 . . . . . 6 (𝑝 = 𝑃 → ((𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ (𝑃𝑞 ∧ (𝑃Line𝑄) = (𝑃Line𝑞))))
9 neeq2 2853 . . . . . . 7 (𝑞 = 𝑄 → (𝑃𝑞𝑃𝑄))
10 oveq2 6623 . . . . . . . 8 (𝑞 = 𝑄 → (𝑃Line𝑞) = (𝑃Line𝑄))
1110eqeq2d 2631 . . . . . . 7 (𝑞 = 𝑄 → ((𝑃Line𝑄) = (𝑃Line𝑞) ↔ (𝑃Line𝑄) = (𝑃Line𝑄)))
129, 11anbi12d 746 . . . . . 6 (𝑞 = 𝑄 → ((𝑃𝑞 ∧ (𝑃Line𝑄) = (𝑃Line𝑞)) ↔ (𝑃𝑄 ∧ (𝑃Line𝑄) = (𝑃Line𝑄))))
138, 12rspc2ev 3313 . . . . 5 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ (𝑃𝑄 ∧ (𝑃Line𝑄) = (𝑃Line𝑄))) → ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))
141, 2, 3, 4, 13syl112anc 1327 . . . 4 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) → ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))
15 fveq2 6158 . . . . . 6 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
1615rexeqdv 3138 . . . . . 6 (𝑛 = 𝑁 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑁)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))))
1715, 16rexeqbidv 3146 . . . . 5 (𝑛 = 𝑁 → (∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))))
1817rspcev 3299 . . . 4 ((𝑁 ∈ ℕ ∧ ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) → ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))
1914, 18sylan2 491 . . 3 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))
20 ellines 31954 . . 3 ((𝑃Line𝑄) ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))
2119, 20sylibr 224 . 2 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → (𝑃Line𝑄) ∈ LinesEE)
22 linerflx1 31951 . 2 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → 𝑃 ∈ (𝑃Line𝑄))
23 linerflx2 31953 . 2 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → 𝑄 ∈ (𝑃Line𝑄))
24 eleq2 2687 . . . 4 (𝑥 = (𝑃Line𝑄) → (𝑃𝑥𝑃 ∈ (𝑃Line𝑄)))
25 eleq2 2687 . . . 4 (𝑥 = (𝑃Line𝑄) → (𝑄𝑥𝑄 ∈ (𝑃Line𝑄)))
2624, 25anbi12d 746 . . 3 (𝑥 = (𝑃Line𝑄) → ((𝑃𝑥𝑄𝑥) ↔ (𝑃 ∈ (𝑃Line𝑄) ∧ 𝑄 ∈ (𝑃Line𝑄))))
2726rspcev 3299 . 2 (((𝑃Line𝑄) ∈ LinesEE ∧ (𝑃 ∈ (𝑃Line𝑄) ∧ 𝑄 ∈ (𝑃Line𝑄))) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
2821, 22, 23, 27syl12anc 1321 1 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909  cfv 5857  (class class class)co 6615  cn 10980  𝔼cee 25702  Linecline2 31936  LinesEEclines2 31938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-ec 7704  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-ee 25705  df-btwn 25706  df-cgr 25707  df-colinear 31841  df-line2 31939  df-lines2 31941
This theorem is referenced by:  linethrueu  31958
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