MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordelssne Unicode version

Theorem ordelssne 4421
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 4408 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 tz7.7 4420 . . 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 1686    =/= wne 2448    C_ wss 3154   Tr wtr 4115   Ord word 4393
This theorem is referenced by:  ordelpss  4422  onelpss  4434  orduniorsuc  4623  ominf  7077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397
  Copyright terms: Public domain W3C validator