Theorem djune 7144

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 6490	. . . . 5 |- 1o =/= (/)
2	1	nesymi 2413	. . . 4 |- -. (/) = 1o
3		1stinl 7140	. . . . 5 |- ( A e. V -> ( 1st ` (inl ` A ) ) = (/) )
4		1stinr 7142	. . . . 5 |- ( B e. W -> ( 1st ` (inr ` B ) ) = 1o )
5	3, 4	eqeqan12d 2212	. . . 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 5558	. . 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 2374	1 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) =/= (inr ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   (/)c0 3450   ` cfv 5258   1stc1st 6196   1oc1o 6467  inlcinl 7111  inrcinr 7112

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-1o 6474  df-inl 7113  df-inr 7114

This theorem is referenced by:  omp1eomlem  7160  difinfsnlem  7165  difinfsn  7166  fodjuomnilemdc  7210  exmidfodomrlemr  7269  exmidfodomrlemrALT  7270