MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssrelrel Unicode version

Theorem ssrelrel 5108
Description: A subclass relationship determined by ordered triples. Use relrelss 5536 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ssrelrel
Distinct variable groups:   , , ,   , , ,

Proof of Theorem ssrelrel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3497 . . . 4
21alrimiv 1719 . . 3
32alrimivv 1720 . 2
4 elvvv 5064 . . . . . . . 8
5 eleq1 2529 . . . . . . . . . . . . . 14
6 eleq1 2529 . . . . . . . . . . . . . 14
75, 6imbi12d 320 . . . . . . . . . . . . 13
87biimprcd 225 . . . . . . . . . . . 12
98alimi 1633 . . . . . . . . . . 11
10 19.23v 1760 . . . . . . . . . . 11
119, 10sylib 196 . . . . . . . . . 10
12112alimi 1634 . . . . . . . . 9
13 19.23vv 1761 . . . . . . . . 9
1412, 13sylib 196 . . . . . . . 8
154, 14syl5bi 217 . . . . . . 7
1615com23 78 . . . . . 6
1716a2d 26 . . . . 5
1817alimdv 1709 . . . 4
19 dfss2 3492 . . . 4
20 dfss2 3492 . . . 4
2118, 19, 203imtr4g 270 . . 3
2221com12 31 . 2
233, 22impbid2 204 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  C_wss 3475  <.cop 4035  X.cxp 5002
This theorem is referenced by:  eqrelrel  5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010
  Copyright terms: Public domain W3C validator