Theorem ordelss 6334

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 6332	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 5203	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	2	imp 406	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
4	1, 3	sylan 581	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ⊆ wss 3890  Tr wtr 5193  Ord word 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3432  df-ss 3907  df-uni 4852  df-tr 5194  df-ord 6321

This theorem is referenced by:  onfr  6357  onelss  6360  ordtri2or2  6419  onfununi  8275  smores3  8287  tfrlem1  8309  tfrlem9a  8319  tz7.44-2  8340  tz7.44-3  8341  oaabslem  8577  oaabs2  8579  omabslem  8580  omabs  8581  findcard3  9187  nnsdomg  9203  ordiso2  9424  ordtypelem2  9428  ordtypelem6  9432  ordtypelem7  9433  cantnf  9608  cnfcomlem  9614  ttrcltr  9631  cardmin2  9917  infxpenlem  9929  iunfictbso  10030  dfac12lem2  10061  dfac12lem3  10062  unctb  10120  ackbij2lem1  10134  ackbij1lem3  10137  ackbij1lem18  10152  ackbij2  10158  ttukeylem6  10430  ttukeylem7  10431  alephexp1  10496  fpwwe2lem7  10554  pwfseqlem3  10577  pwdjundom  10584  fz1isolem  14417  noinfbday  27701  onsuct0  36642  finxpreclem4  37727  nadd2rabtr  43833  grur1cld  44680