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Theorem bnj529 28770
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj529.1  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj529  |-  ( M  e.  D  ->  (/)  e.  M
)

Proof of Theorem bnj529
StepHypRef Expression
1 eldifsn 3749 . . . 4  |-  ( M  e.  ( om  \  { (/)
} )  <->  ( M  e.  om  /\  M  =/=  (/) ) )
21biimpi 186 . . 3  |-  ( M  e.  ( om  \  { (/)
} )  ->  ( M  e.  om  /\  M  =/=  (/) ) )
3 bnj529.1 . . 3  |-  D  =  ( om  \  { (/)
} )
42, 3eleq2s 2375 . 2  |-  ( M  e.  D  ->  ( M  e.  om  /\  M  =/=  (/) ) )
5 nnord 4664 . . 3  |-  ( M  e.  om  ->  Ord  M )
65anim1i 551 . 2  |-  ( ( M  e.  om  /\  M  =/=  (/) )  ->  ( Ord  M  /\  M  =/=  (/) ) )
7 ord0eln0 4446 . . 3  |-  ( Ord 
M  ->  ( (/)  e.  M  <->  M  =/=  (/) ) )
87biimpar 471 . 2  |-  ( ( Ord  M  /\  M  =/=  (/) )  ->  (/)  e.  M
)
94, 6, 83syl 18 1  |-  ( M  e.  D  ->  (/)  e.  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   (/)c0 3455   {csn 3640   Ord word 4391   omcom 4656
This theorem is referenced by:  bnj545  28927  bnj900  28961  bnj929  28968
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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