Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssrelf Structured version   Unicode version

Theorem ssrelf 27609
 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.) (Revised by Thierry Arnoux, 6-Nov-2017.)
Hypotheses
Ref Expression
eqrelrd2.1
eqrelrd2.2
eqrelrd2.3
eqrelrd2.4
eqrelrd2.5
eqrelrd2.6
Assertion
Ref Expression
ssrelf
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem ssrelf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqrelrd2.3 . . . 4
2 eqrelrd2.5 . . . 4
31, 2nfss 3492 . . 3
4 eqrelrd2.4 . . . . 5
5 eqrelrd2.6 . . . . 5
64, 5nfss 3492 . . . 4
7 ssel 3493 . . . 4
86, 7alrimi 1878 . . 3
93, 8alrimi 1878 . 2
10 eleq1 2529 . . . . . . . . . . 11
11 eleq1 2529 . . . . . . . . . . 11
1210, 11imbi12d 320 . . . . . . . . . 10
1312biimprcd 225 . . . . . . . . 9
14132alimi 1635 . . . . . . . 8
154nfcri 2612 . . . . . . . . . . . 12
165nfcri 2612 . . . . . . . . . . . 12
1715, 16nfim 1921 . . . . . . . . . . 11
181719.23 1911 . . . . . . . . . 10
1918albii 1641 . . . . . . . . 9
201nfcri 2612 . . . . . . . . . . 11
212nfcri 2612 . . . . . . . . . . 11
2220, 21nfim 1921 . . . . . . . . . 10
232219.23 1911 . . . . . . . . 9
2419, 23bitri 249 . . . . . . . 8
2514, 24sylib 196 . . . . . . 7
2625com23 78 . . . . . 6
2726a2d 26 . . . . 5
2827alimdv 1710 . . . 4
29 df-rel 5015 . . . . 5
30 dfss2 3488 . . . . 5
31 elvv 5067 . . . . . . 7
3231imbi2i 312 . . . . . 6
3332albii 1641 . . . . 5
3429, 30, 333bitri 271 . . . 4
35 dfss2 3488 . . . 4
3628, 34, 353imtr4g 270 . . 3
3736com12 31 . 2
389, 37impbid2 204 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1393   wceq 1395  wex 1613  wnf 1617   wcel 1819  wnfc 2605  cvv 3109   wss 3471  cop 4038   cxp 5006   wrel 5013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014  df-rel 5015 This theorem is referenced by:  eqrelrd2  27610
 Copyright terms: Public domain W3C validator