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Theorem bj-nnen2lp 13141
Description: A version of en2lp 4464 for natural numbers, which does not require ax-setind 4447.

Note: using this theorem and bj-nnelirr 13140, one can remove dependency on ax-setind 4447 from nntri2 6383 and nndcel 6389; 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 13140 . . 3 (𝐵 ∈ ω → ¬ 𝐵𝐵)
21adantl 275 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ 𝐵𝐵)
3 bj-nntrans 13138 . . . . 5 (𝐵 ∈ ω → (𝐴𝐵𝐴𝐵))
43adantl 275 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
5 ssel 3086 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐵𝐵))
64, 5syl6 33 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐵𝐴𝐵𝐵)))
76impd 252 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵𝐵𝐴) → 𝐵𝐵))
82, 7mtod 652 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴𝐵𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1480  wss 3066  ωcom 4499
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdor 13003  ax-bdn 13004  ax-bdal 13005  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071  ax-infvn 13128
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500  df-bdc 13028  df-bj-ind 13114
This theorem is referenced by:  bj-peano4  13142
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