New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ssrel Unicode version

Theorem ssrel 4844
 Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 2-Aug-1994.) (Revised by set.mm contributors, 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 3267 . . 3
21alrimivv 1632 . 2
3 vex 2862 . . . . . . 7
43proj1ex 4593 . . . . . 6 Proj1
5 opeq1 4578 . . . . . . . . 9 Proj1 Proj1
65eleq1d 2419 . . . . . . . 8 Proj1 Proj1
75eleq1d 2419 . . . . . . . 8 Proj1 Proj1
86, 7imbi12d 311 . . . . . . 7 Proj1 Proj1 Proj1
98albidv 1625 . . . . . 6 Proj1 Proj1 Proj1
104, 9spcv 2945 . . . . 5 Proj1 Proj1
113proj2ex 4594 . . . . . 6 Proj2
12 opeq2 4579 . . . . . . . 8 Proj2 Proj1 Proj1 Proj2
1312eleq1d 2419 . . . . . . 7 Proj2 Proj1 Proj1 Proj2
1412eleq1d 2419 . . . . . . 7 Proj2 Proj1 Proj1 Proj2
1513, 14imbi12d 311 . . . . . 6 Proj2 Proj1 Proj1 Proj1 Proj2 Proj1 Proj2
1611, 15spcv 2945 . . . . 5 Proj1 Proj1 Proj1 Proj2 Proj1 Proj2
1710, 16syl 15 . . . 4 Proj1 Proj2 Proj1 Proj2
18 opeq 4619 . . . . 5 Proj1 Proj2
1918eleq1i 2416 . . . 4 Proj1 Proj2
2018eleq1i 2416 . . . 4 Proj1 Proj2
2117, 19, 203imtr4g 261 . . 3
2221ssrdv 3278 . 2
232, 22impbii 180 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176  wal 1540   wceq 1642   wcel 1710   wss 3257  cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568 This theorem is referenced by:  eqrel  4845  ssopr  4846  relssi  4848  relssdv  4849  cotr  5026  cnvsym  5027  intasym  5028  intirr  5029  ssdmrn  5099  dffun2  5119  fvfullfunlem2  5862
 Copyright terms: Public domain W3C validator