Theorem ordelss 4362

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 4361	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4094	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	2	imp 123	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
4	1, 3	sylan 281	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2141   ⊆ wss 3121  Tr wtr 4085  Ord word 4345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349

This theorem is referenced by:  ordelord  4364  onelss  4370  ordsuc  4545  smores3  6269  tfrlem1  6284  tfrlemisucaccv  6301  tfrlemiubacc  6306  tfr1onlemsucaccv  6317  tfr1onlemubacc  6322  tfrcllemsucaccv  6330  tfrcllemubacc  6335  nntri1  6472  nnsseleq  6477  fict  6842  infnfi  6869  isinfinf  6871  ordiso2  7008  hashinfuni  10698