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Theorem onpsssuc 4626
Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
onpsssuc  |-  ( A  e.  On  ->  A  C.  suc  A )

Proof of Theorem onpsssuc
StepHypRef Expression
1 sucidg 4486 . 2  |-  ( A  e.  On  ->  A  e.  suc  A )
2 eloni 4418 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordsuc 4621 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
42, 3sylib 188 . . 3  |-  ( A  e.  On  ->  Ord  suc 
A )
5 ordelpss 4436 . . 3  |-  ( ( Ord  A  /\  Ord  suc 
A )  ->  ( A  e.  suc  A  <->  A  C.  suc  A ) )
62, 4, 5syl2anc 642 . 2  |-  ( A  e.  On  ->  ( A  e.  suc  A  <->  A  C.  suc  A ) )
71, 6mpbid 201 1  |-  ( A  e.  On  ->  A  C.  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696    C. wpss 3166   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  ackbij1lem15  7876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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