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Theorem bj-nnen2lp 13079
Description: A version of en2lp 4439 for natural numbers, which does not require ax-setind 4422.

Note: using this theorem and bj-nnelirr 13078, one can remove dependency on ax-setind 4422 from nntri2 6358 and nndcel 6364; 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 13078 . . 3  |-  ( B  e.  om  ->  -.  B  e.  B )
21adantl 275 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  -.  B  e.  B
)
3 bj-nntrans 13076 . . . . 5  |-  ( B  e.  om  ->  ( A  e.  B  ->  A 
C_  B ) )
43adantl 275 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 ssel 3061 . . . 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 252 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  e.  B  /\  B  e.  A )  ->  B  e.  B ) )
82, 7mtod 637 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 103    e. wcel 1465    C_ wss 3041   omcom 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdor 12941  ax-bdn 12942  ax-bdal 12943  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bj-peano4  13080
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