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Theorem djune 6929
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 6295 . . . . 5 1o ≠ ∅
21nesymi 2329 . . . 4 ¬ ∅ = 1o
3 1stinl 6925 . . . . 5 (𝐴𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4 1stinr 6927 . . . . 5 (𝐵𝑊 → (1st ‘(inr‘𝐵)) = 1o)
53, 4eqeqan12d 2131 . . . 4 ((𝐴𝑉𝐵𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
62, 5mtbiri 647 . . 3 ((𝐴𝑉𝐵𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7 fveq2 5387 . . 3 ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
86, 7nsyl 600 . 2 ((𝐴𝑉𝐵𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
98neqned 2290 1 ((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  wne 2283  c0 3331  cfv 5091  1st c1st 6002  1oc1o 6272  inlcinl 6896  inrcinr 6897
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-1st 6004  df-1o 6279  df-inl 6898  df-inr 6899
This theorem is referenced by:  omp1eomlem  6945  difinfsnlem  6950  difinfsn  6951  fodjuomnilemdc  6982  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023
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