Theorem onelin 3100

Description: An element of an ordinal number equals the intersection with it.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onelin	|- (B e. A -> B = (B i^i A))

Proof of Theorem onelin

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	onelss 3097	. 2 |- (B e. A -> B (_ A)
3		dfss 2052	. 2 |- (B (_ A <-> B = (B i^i A))
4	2, 3	sylib 198	1 |- (B e. A -> B = (B i^i A))

Syntax hints:   -> wi 3   = wceq 955   e. wcel 957   i^i cin 2044   (_ wss 2045  Oncon0 2945

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-tr 2678  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949

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