Theorem djune 7139

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 6487	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2410	. . . 4 |- -. (/) = 1o
3		1stinl 7135	. . . . 5 |- ( A e. V -> ( 1st ` (inl ` A ) ) = (/) )
4		1stinr 7137	. . . . 5 |- ( B e. W -> ( 1st ` (inr ` B ) ) = 1o )
5	3, 4	eqeqan12d 2209	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) <-> (/) = 1o ) )
6	2, 5	mtbiri 676	. . 3 |- ( ( A e. V /\ B e. W ) -> -. ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) )
7		fveq2 5555	. . 3 |- ( (inl ` A ) = (inr ` B ) -> ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) )
8	6, 7	nsyl 629	. 2 |- ( ( A e. V /\ B e. W ) -> -. (inl ` A ) = (inr ` B ) )
9	8	neqned 2371	1 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) =/= (inr ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   (/)c0 3447   ` cfv 5255   1stc1st 6193   1oc1o 6464  inlcinl 7106  inrcinr 7107

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-1st 6195  df-1o 6471  df-inl 7108  df-inr 7109

This theorem is referenced by:  omp1eomlem  7155  difinfsnlem  7160  difinfsn  7161  fodjuomnilemdc  7205  exmidfodomrlemr  7264  exmidfodomrlemrALT  7265