Theorem onelsst 2995

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 2953	. 2 |- (A e. On -> Ord A)
2		ordtr 2957	. 2 |- (Ord A -> Tr A)
3		trss 2684	. 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 956   (_ wss 2043  Tr wtr 2675  Ord word 2942  Oncon0 2943

This theorem is referenced by:  ordunidif 3000  suceloni 3057  onelss 3095  snsn0non 3120  tfrlem1 3902  tfrlem5 3906  tfrlem9 3910  tfrlem11 3912  oaordex 4182  oaass 4185  odi 4200  omass 4201  oewordri 4209  ondomon 4836  cfub 4888  cfsuc 4895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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