Theorem onelss 3106

Description: A member of an ordinal number is a subset of it.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onelss	|- (B e. A -> B (_ A)

Proof of Theorem onelss

Step	Hyp	Ref	Expression
1		on.1	. 2 |- A e. On
2		onelsst 3006	. 2 |- (A e. On -> (B e. A -> B (_ A))
3	1, 2	ax-mp 7	1 |- (B e. A -> B (_ A)

Syntax hints:   -> wi 3   e. wcel 960   (_ wss 2050  Oncon0 2954

This theorem is referenced by:  onelin 3109  onelun 3110  oawordeulem 4194  carddom 4846  cardsdomel 4863  cardaleph 4896

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  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-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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