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Theorem ssrel 5096
Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ssrel
Distinct variable groups:   , ,   , ,

Proof of Theorem ssrel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3497 . . 3
21alrimivv 1720 . 2
3 eleq1 2529 . . . . . . . . . . 11
4 eleq1 2529 . . . . . . . . . . 11
53, 4imbi12d 320 . . . . . . . . . 10
65biimprcd 225 . . . . . . . . 9
762alimi 1634 . . . . . . . 8
8 19.23vv 1761 . . . . . . . 8
97, 8sylib 196 . . . . . . 7
109com23 78 . . . . . 6
1110a2d 26 . . . . 5
1211alimdv 1709 . . . 4
13 df-rel 5011 . . . . 5
14 dfss2 3492 . . . . 5
15 elvv 5063 . . . . . . 7
1615imbi2i 312 . . . . . 6
1716albii 1640 . . . . 5
1813, 14, 173bitri 271 . . . 4
19 dfss2 3492 . . . 4
2012, 18, 193imtr4g 270 . . 3
2120com12 31 . 2
222, 21impbid2 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  Relwrel 5009
This theorem is referenced by:  eqrel  5097  relssi  5099  relssdv  5100  cotrg  5383  cnvsym  5386  intasym  5387  intirr  5390  codir  5392  qfto  5393  dfso2  29183  dfpo2  29184  dffun10  29564  imagesset  29603
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-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  df-rel 5011
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