Theorem onelsst 3006

Description: An element of an ordinal number is a subset of the number.

Assertion

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

Proof of Theorem onelsst

Step	Hyp	Ref	Expression
1		eloni 2964	. 2 |- (A e. On -> Ord A)
2		ordtr 2968	. 2 |- (Ord A -> Tr A)
3		trss 2694	. 2 |- (Tr A -> (B e. A -> B (_ A))
4	1, 2, 3	3syl 20	1 |- (A e. On -> (B e. A -> B (_ A))

Syntax hints:   -> wi 3   e. wcel 960   (_ wss 2050  Tr wtr 2685  Ord word 2953  Oncon0 2954

This theorem is referenced by:  ordunidif 3011  suceloni 3068  onelss 3106  snsn0non 3131  tfrlem1 3917  tfrlem5 3921  tfrlem9 3925  tfrlem11 3927  oaordex 4198  oaass 4201  odi 4216  omass 4217  oewordri 4225  ondomon 4867  cfub 4920  cfsuc 4927

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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