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Theorem bj-nnen2lp 15446
Description: A version of en2lp 4586 for natural numbers, which does not require ax-setind 4569.

Note: using this theorem and bj-nnelirr 15445, one can remove dependency on ax-setind 4569 from nntri2 6547 and nndcel 6553; 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 15445 . . 3  |-  ( B  e.  om  ->  -.  B  e.  B )
21adantl 277 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  B  e.  B
)
3 bj-nntrans 15443 . . . . 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 3173 . . . 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 664 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 2164    C_ wss 3153   omcom 4622
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdn 15309  ax-bdal 15310  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376  ax-infvn 15433
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623  df-bdc 15333  df-bj-ind 15419
This theorem is referenced by:  bj-peano4  15447
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