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Theorem om00el 7143
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 7142 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
21necon3abid 2628 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =/=  (/)  <->  -.  ( A  =  (/)  \/  B  =  (/) ) ) )
3 omcl 7104 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
4 on0eln0 4847 . . 3  |-  ( ( A  .o  B )  e.  On  ->  ( (/) 
e.  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
53, 4syl 16 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( A  .o  B )  =/=  (/) ) )
6 on0eln0 4847 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7 on0eln0 4847 . . . 4  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
86, 7bi2anan9 871 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  <-> 
( A  =/=  (/)  /\  B  =/=  (/) ) ) )
9 neanior 2707 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
108, 9syl6bb 261 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) ) )
112, 5, 103bitr4d 285 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  ( A  .o  B )  <->  ( (/)  e.  A  /\  (/)  e.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   (/)c0 3711   Oncon0 4792  (class class class)co 6196    .o comu 7046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-omul 7053
This theorem is referenced by:  odi  7146  oeoe  7166  omxpenlem  7537
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