Theorem ordelss 4414

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 4413	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4140	. . 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 2167   ⊆ wss 3157  Tr wtr 4131  Ord word 4397

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401

This theorem is referenced by:  ordelord  4416  onelss  4422  ordsuc  4599  smores3  6351  tfrlem1  6366  tfrlemisucaccv  6383  tfrlemiubacc  6388  tfr1onlemsucaccv  6399  tfr1onlemubacc  6404  tfrcllemsucaccv  6412  tfrcllemubacc  6417  nntri1  6554  nnsseleq  6559  fict  6929  infnfi  6956  isinfinf  6958  ordiso2  7101  hashinfuni  10869