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Theorem bj-nnen2lp 11034
Description: A version of en2lp 4325 for natural numbers, which does not require ax-setind 4308.

Note: using this theorem and bj-nnelirr 11033, one can remove dependency on ax-setind 4308 from nntri2 6159 and nndcel 6165; 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) → ¬ (𝐴𝐵𝐵𝐴))

Proof of Theorem bj-nnen2lp
StepHypRef Expression
1 bj-nnelirr 11033 . . 3 (𝐵 ∈ ω → ¬ 𝐵𝐵)
21adantl 271 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ 𝐵𝐵)
3 bj-nntrans 11031 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴𝐵))
43adantl 271 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
5 ssel 3002 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
64, 5syl6 33 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐵𝐴𝐵𝐵)))
76impd 251 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵𝐵𝐴) → 𝐵𝐵))
82, 7mtod 622 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wcel 1434  wss 2982  ωcom 4359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3924  ax-pr 3992  ax-un 4216  ax-bd0 10889  ax-bdor 10892  ax-bdn 10893  ax-bdal 10894  ax-bdex 10895  ax-bdeq 10896  ax-bdel 10897  ax-bdsb 10898  ax-bdsep 10960  ax-infvn 11021
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423  df-uni 3622  df-int 3657  df-suc 4154  df-iom 4360  df-bdc 10917  df-bj-ind 11007
This theorem is referenced by:  bj-peano4  11035
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