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Theorem elprneb 38857
 Description: An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.)
Assertion
Ref Expression
elprneb

Proof of Theorem elprneb
StepHypRef Expression
1 elpri 3976 . . 3
2 neeq1 2705 . . . . . 6
32eqcoms 2479 . . . . 5
4 pm5.1 875 . . . . . 6
54ex 441 . . . . 5
63, 5sylbid 223 . . . 4
7 neeq2 2706 . . . . 5
8 nesym 2699 . . . . . . . 8
9 pm5.1 875 . . . . . . . 8
108, 9sylan2b 483 . . . . . . 7
1110necon2abid 2685 . . . . . 6
1211ex 441 . . . . 5
137, 12sylbird 243 . . . 4
146, 13jaoi 386 . . 3
151, 14syl 17 . 2
1615imp 436 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   wceq 1452   wcel 1904   wne 2641  cpr 3961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962 This theorem is referenced by:  dfodd5  38934
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