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Theorem invdisj 3759
 Description: If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
invdisj Disj
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem invdisj
StepHypRef Expression
1 nfra2xy 2364 . . 3
2 df-ral 2311 . . . . 5
3 rsp 2369 . . . . . . . . 9
4 eqcom 2042 . . . . . . . . 9
53, 4syl6ib 150 . . . . . . . 8
65imim2i 12 . . . . . . 7
76impd 242 . . . . . 6
87alimi 1344 . . . . 5
92, 8sylbi 114 . . . 4
10 mo2icl 2720 . . . 4
119, 10syl 14 . . 3
121, 11alrimi 1415 . 2
13 dfdisj2 3747 . 2 Disj
1412, 13sylibr 137 1 Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97  wal 1241   wceq 1243   wcel 1393  wmo 1901  wral 2306  Disj wdisj 3745 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rmo 2314  df-v 2559  df-disj 3746 This theorem is referenced by: (None)
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