Theorem onel 3104

Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onel	|- (B e. A -> B e. On)

Proof of Theorem onel

Step	Hyp	Ref	Expression
1		on.1	. 2 |- A e. On
2		onelon 2978	. 2 |- ((A e. On /\ B e. A) -> B e. On)
3	1, 2	mpan 697	1 |- (B e. A -> B e. On)

Syntax hints:   -> wi 3   e. wcel 960  Oncon0 2954

This theorem is referenced by:  onssneli 3107  oawordeulem 4194  rankr1 4684  rankuni 4708  cardne 4840  cardval2 4866  alephval2 4913

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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