Theorem limelon 2286

Description: A limit ordinal class that is also a set is an ordinal number.

Assertion

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

Proof of Theorem limelon

Step	Hyp	Ref	Expression
1		elong 2207	. . 3 ⊢ (A ∈ B → (A ∈ On ↔ Ord A))
2		limord 2283	. . 3 ⊢ (Lim A → Ord A)
3	1, 2	syl5bir 184	. 2 ⊢ (A ∈ B → (Lim A → A ∈ On))
4	3	imp 277	1 ⊢ ((A ∈ B ∧ Lim A) → A ∈ On)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Ord word 2198  Oncon0 2199  Lim wlim 2200

This theorem is referenced by:  dfom2 2374  tfindsg2 2403  rdglimt 2986  oalim 3135  omlim 3136  oelim 3137  oalimcl 3162  oaass 3163  oen0 3165  r1pwcl 3530  alephordi 3679  cflim 3704

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204

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