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Theorem djune 7022
Description: Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
Assertion
Ref Expression
djune ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Proof of Theorem djune
StepHypRef Expression
1 1n0 6379 . . . . 5 1o ≠ ∅
21nesymi 2373 . . . 4 ¬ ∅ = 1o
3 1stinl 7018 . . . . 5 (𝐴𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4 1stinr 7020 . . . . 5 (𝐵𝑊 → (1st ‘(inr‘𝐵)) = 1o)
53, 4eqeqan12d 2173 . . . 4 ((𝐴𝑉𝐵𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
62, 5mtbiri 665 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7 fveq2 5468 . . 3 ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
86, 7nsyl 618 . 2 ((𝐴𝑉𝐵𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
98neqned 2334 1 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wne 2327  c0 3394  cfv 5170  1st c1st 6086  1oc1o 6356  inlcinl 6989  inrcinr 6990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fv 5178  df-1st 6088  df-1o 6363  df-inl 6991  df-inr 6992
This theorem is referenced by:  omp1eomlem  7038  difinfsnlem  7043  difinfsn  7044  fodjuomnilemdc  7087  exmidfodomrlemr  7137  exmidfodomrlemrALT  7138
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