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Theorem nntri2 5988
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
Assertion
Ref Expression
nntri2 ((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))

Proof of Theorem nntri2
StepHypRef Expression
1 elirr 4208 . . . . 5 ¬ A A
2 eleq2 2083 . . . . 5 (A = B → (A AA B))
31, 2mtbii 586 . . . 4 (A = B → ¬ A B)
43con2i 545 . . 3 (A B → ¬ A = B)
5 en2lp 4216 . . . 4 ¬ (A B B A)
65imnani 612 . . 3 (A B → ¬ B A)
7 ioran 656 . . 3 (¬ (A = B B A) ↔ (¬ A = B ¬ B A))
84, 6, 7sylanbrc 396 . 2 (A B → ¬ (A = B B A))
9 nntri3or 5987 . . . . 5 ((A 𝜔 B 𝜔) → (A B A = B B A))
10 3orass 876 . . . . 5 ((A B A = B B A) ↔ (A B (A = B B A)))
119, 10sylib 127 . . . 4 ((A 𝜔 B 𝜔) → (A B (A = B B A)))
1211orcomd 635 . . 3 ((A 𝜔 B 𝜔) → ((A = B B A) A B))
1312ord 630 . 2 ((A 𝜔 B 𝜔) → (¬ (A = B B A) → A B))
148, 13impbid2 131 1 ((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 616   w3o 872   = wceq 1228   wcel 1374  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-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241
This theorem is referenced by:  nnaord  5993  nnmord  6001  pitric  6181
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