Theorem ordelon 4415

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 4413	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
2		elong 4405	. . 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 2164  Ord word 4394  Oncon0 4395

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129  df-iord 4398  df-on 4400

This theorem is referenced by:  onelon  4416  ordsson  4525  ordpwsucss  4600  tfr1onlemsucfn  6395  tfr1onlemsucaccv  6396  tfr1onlembfn  6399  tfr1onlemubacc  6401  tfr1onlemaccex  6403  tfrcllemsucfn  6408  tfrcllemsucaccv  6409  tfrcllembfn  6412  tfrcllemubacc  6414  tfrcllemaccex  6416  tfrcl  6419