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Theorem bnj529 29171
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 3929 . . . 4  |-  ( M  e.  ( om  \  { (/)
} )  <->  ( M  e.  om  /\  M  =/=  (/) ) )
21biimpi 188 . . 3  |-  ( M  e.  ( om  \  { (/)
} )  ->  ( M  e.  om  /\  M  =/=  (/) ) )
3 bnj529.1 . . 3  |-  D  =  ( om  \  { (/)
} )
42, 3eleq2s 2530 . 2  |-  ( M  e.  D  ->  ( M  e.  om  /\  M  =/=  (/) ) )
5 nnord 4855 . . 3  |-  ( M  e.  om  ->  Ord  M )
65anim1i 553 . 2  |-  ( ( M  e.  om  /\  M  =/=  (/) )  ->  ( Ord  M  /\  M  =/=  (/) ) )
7 ord0eln0 4637 . . 3  |-  ( Ord 
M  ->  ( (/)  e.  M  <->  M  =/=  (/) ) )
87biimpar 473 . 2  |-  ( ( Ord  M  /\  M  =/=  (/) )  ->  (/)  e.  M
)
94, 6, 83syl 19 1  |-  ( M  e.  D  ->  (/)  e.  M
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319   (/)c0 3630   {csn 3816   Ord word 4582   omcom 4847
This theorem is referenced by:  bnj545  29328  bnj900  29362  bnj929  29369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848
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