Theorem onelpsst 2995

Description: Relationship between membership and proper subset of an ordinal number.

Assertion

Ref	Expression
onelpsst	|- ((A e. On /\ B e. On) -> (A e. B <-> (A (_ B /\ A =/= B)))

Proof of Theorem onelpsst

Step	Hyp	Ref	Expression
1		ordelssne 2971	. 2 |- ((Ord A /\ Ord B) -> (A e. B <-> (A (_ B /\ A =/= B)))
2		eloni 2955	. 2 |- (A e. On -> Ord A)
3		eloni 2955	. 2 |- (B e. On -> Ord B)
4	1, 2, 3	syl2an 454	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (A (_ B /\ A =/= B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 957   =/= wne 1584   (_ wss 2045  Ord word 2944  Oncon0 2945

This theorem is referenced by:  findsg 3154  tfindsg 3159  oancom 4620  cardsdom 4824  alephord 4862

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-11 966  ax-12 967  ax-13 968  ax-14 969  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  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  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-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949

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