Theorem elon2 2210

Description: An ordinal number is an ordinal set.

Assertion

Ref	Expression
elon2	|- (A e. On <-> (Ord A /\ A e. V))

Proof of Theorem elon2

Step	Hyp	Ref	Expression
1		eloni 2209	. . 3 |- (A e. On -> Ord A)
2		elisset 1354	. . 3 |- (A e. On -> A e. V)
3	1, 2	jca 236	. 2 |- (A e. On -> (Ord A /\ A e. V))
4		elong 2207	. . 3 |- (A e. V -> (A e. On <-> Ord A))
5	4	biimparc 327	. 2 |- ((Ord A /\ A e. V) -> A e. On)
6	3, 5	impbi 139	1 |- (A e. On <-> (Ord A /\ A e. V))

Syntax hints:   <-> wb 127   /\ wa 196   e. wcel 1092  Vcvv 1348  Ord word 2198  Oncon0 2199

This theorem is referenced by:  sucelon 2319

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

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