Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordelss Structured version   Visualization version   GIF version

Theorem ordelss 5727
 Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
Assertion
Ref Expression
ordelss ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)

Proof of Theorem ordelss
StepHypRef Expression
1 ordtr 5725 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 4752 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 445 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 488 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1988   ⊆ wss 3567  Tr wtr 4743  Ord word 5710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-tr 4744  df-ord 5714 This theorem is referenced by:  onfr  5751  onelss  5754  ordtri2or2  5811  onfununi  7423  smores3  7435  tfrlem1  7457  tfrlem9a  7467  tz7.44-2  7488  tz7.44-3  7489  oaabslem  7708  oaabs2  7710  omabslem  7711  omabs  7712  findcard3  8188  nnsdomg  8204  ordiso2  8405  ordtypelem2  8409  ordtypelem6  8413  ordtypelem7  8414  cantnf  8575  cnfcomlem  8581  cardmin2  8809  infxpenlem  8821  iunfictbso  8922  dfac12lem2  8951  dfac12lem3  8952  unctb  9012  ackbij2lem1  9026  ackbij1lem3  9029  ackbij1lem18  9044  ackbij2  9050  ttukeylem6  9321  ttukeylem7  9322  alephexp1  9386  fpwwe2lem8  9444  pwfseqlem3  9467  pwcdandom  9474  fz1isolem  13228  onsuct0  32415  finxpreclem4  33202
 Copyright terms: Public domain W3C validator