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

Step	Hyp	Ref	Expression
1		1n0 6197	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2301	. . . 4 |- -. (/) = 1o
3		1stinl 6763	. . . . 5 |- ( A e. V -> ( 1st ` (inl ` A ) ) = (/) )
4		1stinr 6765	. . . . 5 |- ( B e. W -> ( 1st ` (inr ` B ) ) = 1o )
5	3, 4	eqeqan12d 2103	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( 1st ` (inl ` A ) ) = ( 1st ` (inr ` B ) ) <-> (/) = 1o ) )
6	2, 5	mtbiri 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 ) ) )
8	6, 7	nsyl 593	. 2 |- ( ( A e. V /\ B e. W ) -> -. (inl ` A ) = (inr ` B ) )
9	8	neqned 2262	1 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) =/= (inr ` B ) )

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