Theorem djune 7418

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

Step	Hyp	Ref	Expression
1		1n0 6705	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2466	. . . 4 |- -. (/) = 1o
3		1stinl 7414	. . . . 5 |- ( A e. V -> ( 1st ` (inl ` A ) ) = (/) )
4		1stinr 7416	. . . . 5 |- ( B e. W -> ( 1st ` (inr ` B ) ) = 1o )
5	3, 4	eqeqan12d 2254	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) <-> (/) = 1o ) )
6	2, 5	mtbiri 686	. . 3 |- ( ( A e. V /\ B e. W ) -> -. ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) )
7		fveq2 5695	. . 3 |- ( (inl ` A ) = (inr ` B ) -> ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) )
8	6, 7	nsyl 637	. 2 |- ( ( A e. V /\ B e. W ) -> -. (inl ` A ) = (inr ` B ) )
9	8	neqned 2427	1 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) =/= (inr ` B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   ` cfv 5377   1stc1st 6372   1oc1o 6680  inlcinl 7385  inrcinr 7386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374  df-1o 6687  df-inl 7387  df-inr 7388

This theorem is used by:  omp1eomlem  7434  difinfsnlem  7439  difinfsn  7440  fodjuomnilemdc  7484  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555