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Theorem lineintmo 24155
Description: Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
lineintmo  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE  /\  A  =/=  B
)  ->  E* x
( x  e.  A  /\  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem lineintmo
StepHypRef Expression
1 an4 800 . . . . . . 7  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ( y  e.  A  /\  y  e.  B ) )  <->  ( (
x  e.  A  /\  y  e.  A )  /\  ( x  e.  B  /\  y  e.  B
) ) )
2 linethru 24151 . . . . . . . . . . . . 13  |-  ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )  /\  x  =/=  y
)  ->  A  =  ( xLine y ) )
323expa 1156 . . . . . . . . . . . 12  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  x  =/=  y )  ->  A  =  ( xLine y
) )
4 linethru 24151 . . . . . . . . . . . . 13  |-  ( ( B  e. LinesEE  /\  (
x  e.  B  /\  y  e.  B )  /\  x  =/=  y
)  ->  B  =  ( xLine y ) )
543expa 1156 . . . . . . . . . . . 12  |-  ( ( ( B  e. LinesEE  /\  (
x  e.  B  /\  y  e.  B )
)  /\  x  =/=  y )  ->  B  =  ( xLine y
) )
6 eqtr3 2277 . . . . . . . . . . . 12  |-  ( ( A  =  ( xLine y )  /\  B  =  ( xLine y
) )  ->  A  =  B )
73, 5, 6syl2an 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e. LinesEE  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  /\  (
( B  e. LinesEE  /\  (
x  e.  B  /\  y  e.  B )
)  /\  x  =/=  y ) )  ->  A  =  B )
87anandirs 807 . . . . . . . . . 10  |-  ( ( ( ( A  e. LinesEE  /\  ( x  e.  A  /\  y  e.  A
) )  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  /\  x  =/=  y
)  ->  A  =  B )
98ex 425 . . . . . . . . 9  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  ->  ( x  =/=  y  ->  A  =  B ) )
109necon1d 2490 . . . . . . . 8  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  ->  ( A  =/= 
B  ->  x  =  y ) )
1110an4s 802 . . . . . . 7  |-  ( ( ( A  e. LinesEE  /\  B  e. LinesEE )  /\  ( ( x  e.  A  /\  y  e.  A )  /\  ( x  e.  B  /\  y  e.  B
) ) )  -> 
( A  =/=  B  ->  x  =  y ) )
121, 11sylan2b 463 . . . . . 6  |-  ( ( ( A  e. LinesEE  /\  B  e. LinesEE )  /\  ( ( x  e.  A  /\  x  e.  B )  /\  ( y  e.  A  /\  y  e.  B
) ) )  -> 
( A  =/=  B  ->  x  =  y ) )
1312ex 425 . . . . 5  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE )  ->  ( ( ( x  e.  A  /\  x  e.  B )  /\  ( y  e.  A  /\  y  e.  B
) )  ->  ( A  =/=  B  ->  x  =  y ) ) )
1413com23 74 . . . 4  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE )  ->  ( A  =/= 
B  ->  ( (
( x  e.  A  /\  x  e.  B
)  /\  ( y  e.  A  /\  y  e.  B ) )  ->  x  =  y )
) )
15143impia 1153 . . 3  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE  /\  A  =/=  B
)  ->  ( (
( x  e.  A  /\  x  e.  B
)  /\  ( y  e.  A  /\  y  e.  B ) )  ->  x  =  y )
)
1615alrimivv 2014 . 2  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE  /\  A  =/=  B
)  ->  A. x A. y ( ( ( x  e.  A  /\  x  e.  B )  /\  ( y  e.  A  /\  y  e.  B
) )  ->  x  =  y ) )
17 eleq1 2318 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
18 eleq1 2318 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
1917, 18anbi12d 694 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  /\  x  e.  B
)  <->  ( y  e.  A  /\  y  e.  B ) ) )
2019mo4 2151 . 2  |-  ( E* x ( x  e.  A  /\  x  e.  B )  <->  A. x A. y ( ( ( x  e.  A  /\  x  e.  B )  /\  ( y  e.  A  /\  y  e.  B
) )  ->  x  =  y ) )
2116, 20sylibr 205 1  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE  /\  A  =/=  B
)  ->  E* x
( x  e.  A  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   E*wmo 2119    =/= wne 2421  (class class class)co 5792  Linecline2 24132  LinesEEclines2 24134
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-ec 6630  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-ee 23894  df-btwn 23895  df-cgr 23896  df-ofs 23981  df-ifs 24037  df-cgr3 24038  df-colinear 24039  df-fs 24040  df-line2 24135  df-lines2 24137
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