Theorem ordelon 4385

Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
ordelon	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)

Proof of Theorem ordelon

Step	Hyp	Ref	Expression
1		ordelord 4383	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
2		elong 4375	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ On ↔ Ord 𝐵))
3	2	adantl 277	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ On ↔ Ord 𝐵))
4	1, 3	mpbird 167	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  Ord word 4364  Oncon0 4365

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  onelon  4386  ordsson  4493  ordpwsucss  4568  tfr1onlemsucfn  6344  tfr1onlemsucaccv  6345  tfr1onlembfn  6348  tfr1onlemubacc  6350  tfr1onlemaccex  6352  tfrcllemsucfn  6357  tfrcllemsucaccv  6358  tfrcllembfn  6361  tfrcllemubacc  6363  tfrcllemaccex  6365  tfrcl  6368