Theorem ordelss 4482

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 4481	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4201	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	2	imp 124	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
4	1, 3	sylan 283	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202   ⊆ wss 3201  Tr wtr 4192  Ord word 4465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469

This theorem is referenced by:  ordelord  4484  onelss  4490  ordsuc  4667  smores3  6502  tfrlem1  6517  tfrlemisucaccv  6534  tfrlemiubacc  6539  tfr1onlemsucaccv  6550  tfr1onlemubacc  6555  tfrcllemsucaccv  6563  tfrcllemubacc  6568  nntri1  6707  nnsseleq  6712  fict  7098  infnfi  7127  isinfinf  7129  ordiso2  7277  hashinfuni  11085