Theorem limelon 4281

Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Assertion

Ref	Expression
limelon	⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Proof of Theorem limelon

Step	Hyp	Ref	Expression
1		limord 4277	. . 3 ⊢ (Lim 𝐴 → Ord 𝐴)
2		elong 4255	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
3	1, 2	syl5ibr 155	. 2 ⊢ (𝐴 ∈ 𝐵 → (Lim 𝐴 → 𝐴 ∈ On))
4	3	imp 123	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1463  Ord word 4244  Oncon0 4245  Lim wlim 4246

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-in 3043  df-ss 3050  df-uni 3703  df-tr 3987  df-iord 4248  df-on 4250  df-ilim 4251

This theorem is referenced by: (None)