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Theorem ordelssne 4356
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 4343 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 tz7.7 4355 . . 3  |-  ( ( Ord  B  /\  Tr  A )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/= 
B ) ) )
31, 2sylan2 462 . 2  |-  ( ( Ord  B  /\  Ord  A )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
43ancoms 441 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  ( A  C_  B  /\  A  =/=  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    =/= wne 2419    C_ wss 3094   Tr wtr 4053   Ord word 4328
This theorem is referenced by:  ordelpss  4357  onelpss  4369  orduniorsuc  4558  ominf  7008
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332
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