Theorem ordelss 6267

Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Assertion

Ref	Expression
ordelss	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Proof of Theorem ordelss

Step	Hyp	Ref	Expression
1		ordtr 6265	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 5196	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	2	imp 406	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
4	1, 3	sylan 579	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3883  Tr wtr 5187  Ord word 6250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5188  df-ord 6254

This theorem is referenced by:  onfr  6290  onelss  6293  ordtri2or2  6347  onfununi  8143  smores3  8155  tfrlem1  8178  tfrlem9a  8188  tz7.44-2  8209  tz7.44-3  8210  oaabslem  8437  oaabs2  8439  omabslem  8440  omabs  8441  findcard3  8987  nnsdomg  9003  ordiso2  9204  ordtypelem2  9208  ordtypelem6  9212  ordtypelem7  9213  cantnf  9381  cnfcomlem  9387  cardmin2  9688  infxpenlem  9700  iunfictbso  9801  dfac12lem2  9831  dfac12lem3  9832  unctb  9892  ackbij2lem1  9906  ackbij1lem3  9909  ackbij1lem18  9924  ackbij2  9930  ttukeylem6  10201  ttukeylem7  10202  alephexp1  10266  fpwwe2lem7  10324  pwfseqlem3  10347  pwdjundom  10354  fz1isolem  14103  ttrcltr  33702  noinfbday  33850  onsuct0  34557  finxpreclem4  35492  grur1cld  41739