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Theorem ssrel 4931
 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 3310 . . 3
21alrimivv 1639 . 2
3 eleq1 2472 . . . . . . . . . . 11
4 eleq1 2472 . . . . . . . . . . 11
53, 4imbi12d 312 . . . . . . . . . 10
65biimprcd 217 . . . . . . . . 9
762alimi 1566 . . . . . . . 8
8 19.23vv 1911 . . . . . . . 8
97, 8sylib 189 . . . . . . 7
109com23 74 . . . . . 6
1110a2d 24 . . . . 5
1211alimdv 1628 . . . 4
13 df-rel 4852 . . . . 5
14 dfss2 3305 . . . . 5
15 elvv 4903 . . . . . . 7
1615imbi2i 304 . . . . . 6
1716albii 1572 . . . . 5
1813, 14, 173bitri 263 . . . 4
19 dfss2 3305 . . . 4
2012, 18, 193imtr4g 262 . . 3
2120com12 29 . 2
222, 21impbid2 196 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177  wal 1546  wex 1547   wceq 1649   wcel 1721  cvv 2924   wss 3288  cop 3785   cxp 4843   wrel 4850 This theorem is referenced by:  eqrel  4932  relssi  4934  relssdv  4935  cotr  5213  cnvsym  5215  intasym  5216  intirr  5219  codir  5221  qfto  5222  dfso2  25333  dfpo2  25334  dffun10  25675 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-rel 4852
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