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Theorem disj 3262
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj  i^i  (/)
Distinct variable groups:   ,   ,

Proof of Theorem disj
StepHypRef Expression
1 df-in 2918 . . . 4  i^i  {  |  }
21eqeq1i 2044 . . 3  i^i  (/)  {  |  }  (/)
3 abeq1 2144 . . 3  {  |  }  (/)  (/)
4 imnan 623 . . . . 5
5 noel 3222 . . . . . 6  (/)
65nbn 614 . . . . 5  (/)
74, 6bitr2i 174 . . . 4  (/)
87albii 1356 . . 3  (/)
92, 3, 83bitri 195 . 2  i^i  (/)
10 df-ral 2305 . 2
119, 10bitr4i 176 1  i^i  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   {cab 2023  wral 2300    i^i cin 2910   (/)c0 3218
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219
This theorem is referenced by:  disjr  3263  disj1  3264  disjne  3267  renfdisj  6876
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