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Theorem onpsssuc 6627
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 4945 . 2  |-  ( A  e.  On  ->  A  e.  suc  A )
2 eloni 4877 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordsuc 6622 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
42, 3sylib 196 . . 3  |-  ( A  e.  On  ->  Ord  suc 
A )
5 ordelpss 4895 . . 3  |-  ( ( Ord  A  /\  Ord  suc 
A )  ->  ( A  e.  suc  A  <->  A  C.  suc  A
) )
62, 4, 5syl2anc 659 . 2  |-  ( A  e.  On  ->  ( A  e.  suc  A  <->  A  C.  suc  A
) )
71, 6mpbid 210 1  |-  ( A  e.  On  ->  A  C. 
suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1823    C. wpss 3462   Ord word 4866   Oncon0 4867   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873
This theorem is referenced by:  ackbij1lem15  8605
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