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Theorem prneimg 3536
Description: Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
Assertion
Ref Expression
prneimg  U  V  C  X  D  Y  =/=  C  =/=  D  =/=  C  =/=  D  { ,  }  =/=  { C ,  D }

Proof of Theorem prneimg
StepHypRef Expression
1 preq12bg 3535 . . . . 5  U  V  C  X  D  Y  { ,  }  { C ,  D }  C  D  D  C
2 orddi 732 . . . . . 6  C  D  D  C  C  D  C  C  D  D  D  C
3 simpll 481 . . . . . . 7  C  D  C  C  D  D  D  C  C  D
4 pm1.4 645 . . . . . . . 8  D  C  C  D
54ad2antll 460 . . . . . . 7  C  D  C  C  D  D  D  C  C  D
63, 5jca 290 . . . . . 6  C  D  C  C  D  D  D  C  C  D  C  D
72, 6sylbi 114 . . . . 5  C  D  D  C  C  D  C  D
81, 7syl6bi 152 . . . 4  U  V  C  X  D  Y  { ,  }  { C ,  D }  C  D  C  D
9 oranim 806 . . . . . 6  C  D  C  D
10 df-ne 2203 . . . . . . 7  =/=  C  C
11 df-ne 2203 . . . . . . 7  =/=  D  D
1210, 11anbi12i 433 . . . . . 6  =/=  C  =/=  D  C  D
139, 12sylnibr 601 . . . . 5  C  D  =/=  C  =/=  D
14 oranim 806 . . . . . 6  C  D  C  D
15 df-ne 2203 . . . . . . 7  =/=  C  C
16 df-ne 2203 . . . . . . 7  =/=  D  D
1715, 16anbi12i 433 . . . . . 6  =/=  C  =/=  D  C  D
1814, 17sylnibr 601 . . . . 5  C  D  =/=  C  =/=  D
1913, 18anim12i 321 . . . 4  C  D  C  D  =/=  C  =/=  D  =/=  C  =/=  D
208, 19syl6 29 . . 3  U  V  C  X  D  Y  { ,  }  { C ,  D }  =/=  C  =/=  D  =/=  C  =/=  D
21 pm4.56 805 . . 3  =/=  C  =/=  D  =/=  C  =/=  D  =/=  C  =/=  D  =/=  C  =/=  D
2220, 21syl6ib 150 . 2  U  V  C  X  D  Y  { ,  }  { C ,  D }  =/=  C  =/=  D  =/=  C  =/=  D
2322necon2ad 2256 1  U  V  C  X  D  Y  =/=  C  =/=  D  =/=  C  =/=  D  { ,  }  =/=  { C ,  D }
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wo 628   wceq 1242   wcel 1390    =/= wne 2201   {cpr 3368
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-bnd 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
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