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Theorem om00el 6616
Description: The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.)
Assertion
Ref Expression
om00el  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( (/)  e.  A  /\  (/)  e.  B ) ) )

Proof of Theorem om00el
StepHypRef Expression
1 om00 6615 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
21necon3abid 2512 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =/=  (/)  <->  -.  ( A  =  (/)  \/  B  =  (/) ) ) )
3 omcl 6577 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
4 on0eln0 4484 . . 3  |-  ( ( A  .o  B )  e.  On  ->  ( (/) 
e.  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
53, 4syl 15 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
6 on0eln0 4484 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7 on0eln0 4484 . . . 4  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
86, 7bi2anan9 843 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  <-> 
( A  =/=  (/)  /\  B  =/=  (/) ) ) )
9 neanior 2564 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
108, 9syl6bb 252 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) ) )
112, 5, 103bitr4d 276 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( (/)  e.  A  /\  (/)  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   (/)c0 3489   Oncon0 4429  (class class class)co 5900    .o comu 6519
This theorem is referenced by:  odi  6619  oeoe  6639  omxpenlem  7006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-omul 6526
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