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Theorem tpprceq3 4112
 Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
tpprceq3

Proof of Theorem tpprceq3
StepHypRef Expression
1 ianor 491 . 2
2 tprot 4067 . . . 4
3 df-tp 3973 . . . . 5
4 prprc2 4083 . . . . . . 7
54uneq1d 3587 . . . . . 6
6 df-pr 3971 . . . . . . 7
7 prcom 4050 . . . . . . 7
86, 7eqtr3i 2475 . . . . . 6
95, 8syl6eq 2501 . . . . 5
103, 9syl5eq 2497 . . . 4
112, 10syl5eq 2497 . . 3
12 nne 2628 . . . 4
13 tppreq3 4077 . . . . 5
1413eqcoms 2459 . . . 4
1512, 14sylbi 199 . . 3
1611, 15jaoi 381 . 2
171, 16sylbi 199 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 370   wa 371   wceq 1444   wcel 1887   wne 2622  cvv 3045   cun 3402  csn 3968  cpr 3970  ctp 3972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-un 3409  df-nul 3732  df-sn 3969  df-pr 3971  df-tp 3973 This theorem is referenced by:  tppreqb  4113  1to3vfriswmgra  25735
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