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Theorem bj-nnen2lp 10907
Description: A version of en2lp 4305 for natural numbers, which does not require ax-setind 4288.

Note: using this theorem and bj-nnelirr 10906, one can remove dependency on ax-setind 4288 from nntri2 6138 and nndcel 6144; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-nnen2lp  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  ( A  e.  B  /\  B  e.  A ) )

Proof of Theorem bj-nnen2lp
StepHypRef Expression
1 bj-nnelirr 10906 . . 3  |-  ( B  e.  om  ->  -.  B  e.  B )
21adantl 271 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  B  e.  B
)
3 bj-nntrans 10904 . . . . 5  |-  ( B  e.  om  ->  ( A  e.  B  ->  A 
C_  B ) )
43adantl 271 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 ssel 2994 . . . 4  |-  ( A 
C_  B  ->  ( B  e.  A  ->  B  e.  B ) )
64, 5syl6 33 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  ( B  e.  A  ->  B  e.  B ) ) )
76impd 251 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  e.  B  /\  B  e.  A )  ->  B  e.  B ) )
82, 7mtod 622 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434    C_ wss 2974   omcom 4339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10762  ax-bdor 10765  ax-bdn 10766  ax-bdal 10767  ax-bdex 10768  ax-bdeq 10769  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833  ax-infvn 10894
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340  df-bdc 10790  df-bj-ind 10880
This theorem is referenced by:  bj-peano4  10908
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