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Theorem ordelssne 4418
Description: Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)
Assertion
Ref Expression
ordelssne  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )

Proof of Theorem ordelssne
StepHypRef Expression
1 ordtr 4405 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 tz7.7 4417 . . 3  |-  ( ( Ord  B  /\  Tr  A )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/= 
B ) ) )
31, 2sylan2 460 . 2  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
43ancoms 439 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685    =/= wne 2447    C_ wss 3153   Tr wtr 4114   Ord word 4390
This theorem is referenced by:  ordelpss  4419  onelpss  4431  orduniorsuc  4620  ominf  7071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394
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