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Theorem djune 6767
Description: Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
Assertion
Ref Expression
djune  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl `  A )  =/=  (inr `  B )
)

Proof of Theorem djune
StepHypRef Expression
1 1n0 6197 . . . . 5  |-  1o  =/=  (/)
21nesymi 2301 . . . 4  |-  -.  (/)  =  1o
3 1stinl 6763 . . . . 5  |-  ( A  e.  V  ->  ( 1st `  (inl `  A
) )  =  (/) )
4 1stinr 6765 . . . . 5  |-  ( B  e.  W  ->  ( 1st `  (inr `  B
) )  =  1o )
53, 4eqeqan12d 2103 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) )  <->  (/)  =  1o ) )
62, 5mtbiri 635 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) ) )
7 fveq2 5305 . . 3  |-  ( (inl
`  A )  =  (inr `  B )  ->  ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) ) )
86, 7nsyl 593 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  (inl `  A
)  =  (inr `  B ) )
98neqned 2262 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl `  A )  =/=  (inr `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    =/= wne 2255   (/)c0 3286   ` cfv 5015   1stc1st 5909   1oc1o 6174  inlcinl 6735  inrcinr 6736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-1o 6181  df-inl 6737  df-inr 6738
This theorem is referenced by:  fodjuomnilemdc  6797  exmidfodomrlemr  6826  exmidfodomrlemrALT  6827
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