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Theorem lineintmo 24187
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 24183 . . . . . . . . . . . . 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 24183 . . . . . . . . . . . . 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 2303 . . . . . . . . . . . 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 2516 . . . . . . . 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 1623 . 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 2344 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
18 eleq1 2344 . . . 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 2177 . 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 1628    e. wcel 1688   E*wmo 2145    =/= wne 2447  (class class class)co 5819  Linecline2 24164  LinesEEclines2 24166
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-ec 6657  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-sum 12153  df-ee 23926  df-btwn 23927  df-cgr 23928  df-ofs 24013  df-ifs 24069  df-cgr3 24070  df-colinear 24071  df-fs 24072  df-line2 24167  df-lines2 24169
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