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Theorem djune 7108
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 6458 . . . . 5  |-  1o  =/=  (/)
21nesymi 2406 . . . 4  |-  -.  (/)  =  1o
3 1stinl 7104 . . . . 5  |-  ( A  e.  V  ->  ( 1st `  (inl `  A
) )  =  (/) )
4 1stinr 7106 . . . . 5  |-  ( B  e.  W  ->  ( 1st `  (inr `  B
) )  =  1o )
53, 4eqeqan12d 2205 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) )  <->  (/)  =  1o ) )
62, 5mtbiri 676 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) ) )
7 fveq2 5534 . . 3  |-  ( (inl
`  A )  =  (inr `  B )  ->  ( 1st `  (inl `  A ) )  =  ( 1st `  (inr `  B ) ) )
86, 7nsyl 629 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  (inl `  A
)  =  (inr `  B ) )
98neqned 2367 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl `  A )  =/=  (inr `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    =/= wne 2360   (/)c0 3437   ` cfv 5235   1stc1st 6164   1oc1o 6435  inlcinl 7075  inrcinr 7076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fv 5243  df-1st 6166  df-1o 6442  df-inl 7077  df-inr 7078
This theorem is referenced by:  omp1eomlem  7124  difinfsnlem  7129  difinfsn  7130  fodjuomnilemdc  7173  exmidfodomrlemr  7232  exmidfodomrlemrALT  7233
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