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Theorem bj-nnen2lp 7176
Description: A version of en2lp 4216 for natural numbers, which does not require ax-setind 4204.

Note: using this theorem and bj-nnelirr 7175, one can remove dependency on ax-setind 4204 from nntri2 5988 and nndcel 5991; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

Assertion
Ref Expression
bj-nnen2lp ((A 𝜔 B 𝜔) → ¬ (A B B A))

Proof of Theorem bj-nnen2lp
StepHypRef Expression
1 bj-nnelirr 7175 . . 3 (B 𝜔 → ¬ B B)
21adantl 262 . 2 ((A 𝜔 B 𝜔) → ¬ B B)
3 bj-nntrans 7173 . . . . 5 (B 𝜔 → (A BAB))
43adantl 262 . . . 4 ((A 𝜔 B 𝜔) → (A BAB))
5 ssel 2916 . . . 4 (AB → (B AB B))
64, 5syl6 29 . . 3 ((A 𝜔 B 𝜔) → (A B → (B AB B)))
76impd 242 . 2 ((A 𝜔 B 𝜔) → ((A B B A) → B B))
82, 7mtod 576 1 ((A 𝜔 B 𝜔) → ¬ (A B B A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1374  wss 2894  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdn 7044  ax-bdal 7045  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150
This theorem is referenced by:  bj-peano4  7177
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