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Theorem djune 7111
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 6461 . . . . 5 1o ≠ ∅
21nesymi 2406 . . . 4 ¬ ∅ = 1o
3 1stinl 7107 . . . . 5 (𝐴𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4 1stinr 7109 . . . . 5 (𝐵𝑊 → (1st ‘(inr‘𝐵)) = 1o)
53, 4eqeqan12d 2205 . . . 4 ((𝐴𝑉𝐵𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
62, 5mtbiri 676 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7 fveq2 5537 . . 3 ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
86, 7nsyl 629 . 2 ((𝐴𝑉𝐵𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
98neqned 2367 1 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wne 2360  c0 3437  cfv 5238  1st c1st 6167  1oc1o 6438  inlcinl 7078  inrcinr 7079
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-iota 5199  df-fun 5240  df-fv 5246  df-1st 6169  df-1o 6445  df-inl 7080  df-inr 7081
This theorem is referenced by:  omp1eomlem  7127  difinfsnlem  7132  difinfsn  7133  fodjuomnilemdc  7177  exmidfodomrlemr  7236  exmidfodomrlemrALT  7237
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