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Theorem djuinr 6948
 Description: The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 6978 and djufun 6989) while the simpler statement inl inr is easily recovered from it by substituting for both and as done in casefun 6970). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djuinr inl inr

Proof of Theorem djuinr
StepHypRef Expression
1 djulf1or 6941 . . . 4 inl
2 dff1o5 5376 . . . . 5 inl inl inl
32simprbi 273 . . . 4 inl inl
41, 3ax-mp 5 . . 3 inl
5 djurf1or 6942 . . . 4 inr
6 dff1o5 5376 . . . . 5 inr inr inr
76simprbi 273 . . . 4 inr inr
85, 7ax-mp 5 . . 3 inr
94, 8ineq12i 3275 . 2 inl inr
10 1n0 6329 . . . . 5
1110necomi 2393 . . . 4
12 disjsn2 3586 . . . 4
1311, 12ax-mp 5 . . 3
14 xpdisj1 4963 . . 3
1513, 14ax-mp 5 . 2
169, 15eqtri 2160 1 inl inr
 Colors of variables: wff set class Syntax hints:   wceq 1331   wne 2308   cin 3070  c0 3363  csn 3527   cxp 4537   crn 4540   cres 4541  wf1 5120  wf1o 5122  c1o 6306  inlcinl 6930  inrcinr 6931 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-inl 6932  df-inr 6933 This theorem is referenced by:  djuin  6949  casefun  6970  djudom  6978  djufun  6989
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