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Theorem bj-nnen2lp 9414
Description: A version of en2lp 4232 for natural numbers, which does not require ax-setind 4220.

Note: using this theorem and bj-nnelirr 9413, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; 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 9413 . . 3 (B 𝜔 → ¬ B B)
21adantl 262 . 2 ((A 𝜔 B 𝜔) → ¬ B B)
3 bj-nntrans 9411 . . . . 5 (B 𝜔 → (A BAB))
43adantl 262 . . . 4 ((A 𝜔 B 𝜔) → (A BAB))
5 ssel 2933 . . . 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 588 1 ((A 𝜔 B 𝜔) → ¬ (A B B A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wcel 1390  wss 2911  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdor 9271  ax-bdn 9272  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by:  bj-peano4  9415
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