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Theorem otsndisj 4706
 Description: The singletons consisting of ordered triples which have distinct third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Assertion
Ref Expression
otsndisj Disj
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem otsndisj
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 orc 387 . . . . 5
21a1d 26 . . . 4
3 otthg 4685 . . . . . . . . . . . . 13
433expa 1208 . . . . . . . . . . . 12
54adantrr 723 . . . . . . . . . . 11
6 simp3 1010 . . . . . . . . . . 11
75, 6syl6bi 232 . . . . . . . . . 10
87con3rr3 142 . . . . . . . . 9
98imp 431 . . . . . . . 8
109neqned 2631 . . . . . . 7
11 disjsn2 4033 . . . . . . 7
1210, 11syl 17 . . . . . 6
1312olcd 395 . . . . 5
1413ex 436 . . . 4
152, 14pm2.61i 168 . . 3
1615ralrimivva 2809 . 2
17 oteq3 4177 . . . 4
1817sneqd 3980 . . 3
1918disjor 4387 . 2 Disj
2016, 19sylibr 216 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 188   wo 370   wa 371   w3a 985   wceq 1444   wcel 1887   wne 2622  wral 2737   cin 3403  c0 3731  csn 3968  cotp 3976  Disj wdisj 4373 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-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rmo 2745  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-ot 3977  df-disj 4374 This theorem is referenced by:  usgreghash2spotv  25794
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