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Theorem djune 6748
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 6179 . . . . 5 1𝑜 ≠ ∅
21nesymi 2301 . . . 4 ¬ ∅ = 1𝑜
3 1stinl 6744 . . . . 5 (𝐴𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4 1stinr 6746 . . . . 5 (𝐵𝑊 → (1st ‘(inr‘𝐵)) = 1𝑜)
53, 4eqeqan12d 2103 . . . 4 ((𝐴𝑉𝐵𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1𝑜))
62, 5mtbiri 635 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7 fveq2 5289 . . 3 ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
86, 7nsyl 593 . 2 ((𝐴𝑉𝐵𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
98neqned 2262 1 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wne 2255  c0 3284  cfv 5002  1st c1st 5891  1𝑜c1o 6156  inlcinl 6716  inrcinr 6717
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 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-1st 5893  df-1o 6163  df-inl 6718  df-inr 6719
This theorem is referenced by:  fodjuomnilemdc  6778  exmidfodomrlemr  6807  exmidfodomrlemrALT  6808
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