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Theorem ssrelrel 4976
 Description: A subclass relationship determined by ordered triples. Use relrelss 5393 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 3342 . . . 4
21alrimiv 1641 . . 3
32alrimivv 1642 . 2
4 elvvv 4937 . . . . . . . 8
5 eleq1 2496 . . . . . . . . . . . . . 14
6 eleq1 2496 . . . . . . . . . . . . . 14
75, 6imbi12d 312 . . . . . . . . . . . . 13
87biimprcd 217 . . . . . . . . . . . 12
98alimi 1568 . . . . . . . . . . 11
10 19.23v 1914 . . . . . . . . . . 11
119, 10sylib 189 . . . . . . . . . 10
12112alimi 1569 . . . . . . . . 9
13 19.23vv 1915 . . . . . . . . 9
1412, 13sylib 189 . . . . . . . 8
154, 14syl5bi 209 . . . . . . 7
1615com23 74 . . . . . 6
1716a2d 24 . . . . 5
1817alimdv 1631 . . . 4
19 dfss2 3337 . . . 4
20 dfss2 3337 . . . 4
2118, 19, 203imtr4g 262 . . 3
2221com12 29 . 2
233, 22impbid2 196 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177  wal 1549  wex 1550   wceq 1652   wcel 1725  cvv 2956   wss 3320  cop 3817   cxp 4876 This theorem is referenced by:  eqrelrel  4977 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884
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