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Theorem onpsssuc 6535
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 4900 . 2  |-  ( A  e.  On  ->  A  e.  suc  A )
2 eloni 4832 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordsuc 6530 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
42, 3sylib 196 . . 3  |-  ( A  e.  On  ->  Ord  suc 
A )
5 ordelpss 4850 . . 3  |-  ( ( Ord  A  /\  Ord  suc 
A )  ->  ( A  e.  suc  A  <->  A  C.  suc  A
) )
62, 4, 5syl2anc 661 . 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 1758    C. wpss 3432   Ord word 4821   Oncon0 4822   suc csuc 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828
This theorem is referenced by:  ackbij1lem15  8509
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