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Theorem lineintmo 25999
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 an4 798 . . . . . . 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 25995 . . . . . . . . . . . . 13  |-  ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )  /\  x  =/=  y
)  ->  A  =  ( xLine y ) )
323expa 1153 . . . . . . . . . . . 12  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  x  =/=  y )  ->  A  =  ( xLine y
) )
4 linethru 25995 . . . . . . . . . . . . 13  |-  ( ( B  e. LinesEE  /\  (
x  e.  B  /\  y  e.  B )  /\  x  =/=  y
)  ->  B  =  ( xLine y ) )
543expa 1153 . . . . . . . . . . . 12  |-  ( ( ( B  e. LinesEE  /\  (
x  e.  B  /\  y  e.  B )
)  /\  x  =/=  y )  ->  B  =  ( xLine y
) )
6 eqtr3 2427 . . . . . . . . . . . 12  |-  ( ( A  =  ( xLine y )  /\  B  =  ( xLine y
) )  ->  A  =  B )
73, 5, 6syl2an 464 . . . . . . . . . . 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 805 . . . . . . . . . 10  |-  ( ( ( ( A  e. LinesEE  /\  ( x  e.  A  /\  y  e.  A
) )  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  /\  x  =/=  y
)  ->  A  =  B )
98ex 424 . . . . . . . . 9  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  ->  ( x  =/=  y  ->  A  =  B ) )
109necon1d 2640 . . . . . . . 8  |-  ( ( ( A  e. LinesEE  /\  (
x  e.  A  /\  y  e.  A )
)  /\  ( B  e. LinesEE  /\  ( x  e.  B  /\  y  e.  B ) ) )  ->  ( A  =/= 
B  ->  x  =  y ) )
1110an4s 800 . . . . . . 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 462 . . . . . 6  |-  ( ( ( A  e. LinesEE  /\  B  e. LinesEE )  /\  ( ( x  e.  A  /\  x  e.  B )  /\  ( y  e.  A  /\  y  e.  B
) ) )  -> 
( A  =/=  B  ->  x  =  y ) )
1312ex 424 . . . . 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 1150 . . 3  |-  ( ( A  e. LinesEE  /\  B  e. LinesEE  /\  A  =/=  B
)  ->  ( (
( x  e.  A  /\  x  e.  B
)  /\  ( y  e.  A  /\  y  e.  B ) )  ->  x  =  y )
)
1615alrimivv 1639 . 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 2468 . . . 4  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
18 eleq1 2468 . . . 4  |-  ( x  =  y  ->  (
x  e.  B  <->  y  e.  B ) )
1917, 18anbi12d 692 . . 3  |-  ( x  =  y  ->  (
( x  e.  A  /\  x  e.  B
)  <->  ( y  e.  A  /\  y  e.  B ) ) )
2019mo4 2291 . 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 204 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 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   E*wmo 2259    =/= wne 2571  (class class class)co 6044  Linecline2 25976  LinesEEclines2 25978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-ec 6870  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-ee 25738  df-btwn 25739  df-cgr 25740  df-ofs 25825  df-ifs 25881  df-cgr3 25882  df-colinear 25883  df-fs 25884  df-line2 25979  df-lines2 25981
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