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

Theorem ssrelrel 5109
 Description: A subclass relationship determined by ordered triples. Use relrelss 5537 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 3503 . . . 4
21alrimiv 1695 . . 3
32alrimivv 1696 . 2
4 elvvv 5065 . . . . . . . 8
5 eleq1 2539 . . . . . . . . . . . . . 14
6 eleq1 2539 . . . . . . . . . . . . . 14
75, 6imbi12d 320 . . . . . . . . . . . . 13
87biimprcd 225 . . . . . . . . . . . 12
98alimi 1614 . . . . . . . . . . 11
10 19.23v 1932 . . . . . . . . . . 11
119, 10sylib 196 . . . . . . . . . 10
12112alimi 1615 . . . . . . . . 9
13 19.23vv 1934 . . . . . . . . 9
1412, 13sylib 196 . . . . . . . 8
154, 14syl5bi 217 . . . . . . 7
1615com23 78 . . . . . 6
1716a2d 26 . . . . 5
1817alimdv 1685 . . . 4
19 dfss2 3498 . . . 4
20 dfss2 3498 . . . 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  wal 1377   wceq 1379  wex 1596   wcel 1767  cvv 3118   wss 3481  cop 4039   cxp 5003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-xp 5011 This theorem is referenced by:  eqrelrel  5110
 Copyright terms: Public domain W3C validator