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Theorem bj-nnen2lp 14557
Description: A version of en2lp 4552 for natural numbers, which does not require ax-setind 4535.

Note: using this theorem and bj-nnelirr 14556, one can remove dependency on ax-setind 4535 from nntri2 6491 and nndcel 6497; 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 14556 . . 3  |-  ( B  e.  om  ->  -.  B  e.  B )
21adantl 277 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  B  e.  B
)
3 bj-nntrans 14554 . . . . 5  |-  ( B  e.  om  ->  ( A  e.  B  ->  A 
C_  B ) )
43adantl 277 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 ssel 3149 . . . 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 254 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  e.  B  /\  B  e.  A )  ->  B  e.  B ) )
82, 7mtod 663 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 104    e. wcel 2148    C_ wss 3129   omcom 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4128  ax-pr 4208  ax-un 4432  ax-bd0 14416  ax-bdor 14419  ax-bdn 14420  ax-bdal 14421  ax-bdex 14422  ax-bdeq 14423  ax-bdel 14424  ax-bdsb 14425  ax-bdsep 14487  ax-infvn 14544
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4370  df-iom 4589  df-bdc 14444  df-bj-ind 14530
This theorem is referenced by:  bj-peano4  14558
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