ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nntri3 GIF version

Theorem nntri3 6014
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
Assertion
Ref Expression
nntri3 ((A 𝜔 B 𝜔) → (A = B ↔ (¬ A B ¬ B A)))

Proof of Theorem nntri3
StepHypRef Expression
1 elirr 4224 . . . . . 6 ¬ A A
2 eleq2 2098 . . . . . 6 (A = B → (A AA B))
31, 2mtbii 598 . . . . 5 (A = B → ¬ A B)
43con2i 557 . . . 4 (A B → ¬ A = B)
54adantl 262 . . 3 (((A 𝜔 B 𝜔) A B) → ¬ A = B)
6 simpl 102 . . . . 5 ((¬ A B ¬ B A) → ¬ A B)
76con2i 557 . . . 4 (A B → ¬ (¬ A B ¬ B A))
87adantl 262 . . 3 (((A 𝜔 B 𝜔) A B) → ¬ (¬ A B ¬ B A))
95, 82falsed 617 . 2 (((A 𝜔 B 𝜔) A B) → (A = B ↔ (¬ A B ¬ B A)))
10 simpr 103 . . 3 (((A 𝜔 B 𝜔) A = B) → A = B)
11 eleq1 2097 . . . . . 6 (A = B → (A AB A))
121, 11mtbii 598 . . . . 5 (A = B → ¬ B A)
133, 12jca 290 . . . 4 (A = B → (¬ A B ¬ B A))
1413adantl 262 . . 3 (((A 𝜔 B 𝜔) A = B) → (¬ A B ¬ B A))
1510, 142thd 164 . 2 (((A 𝜔 B 𝜔) A = B) → (A = B ↔ (¬ A B ¬ B A)))
1612con2i 557 . . . 4 (B A → ¬ A = B)
1716adantl 262 . . 3 (((A 𝜔 B 𝜔) B A) → ¬ A = B)
18 simpr 103 . . . . 5 ((¬ A B ¬ B A) → ¬ B A)
1918con2i 557 . . . 4 (B A → ¬ (¬ A B ¬ B A))
2019adantl 262 . . 3 (((A 𝜔 B 𝜔) B A) → ¬ (¬ A B ¬ B A))
2117, 202falsed 617 . 2 (((A 𝜔 B 𝜔) B A) → (A = B ↔ (¬ A B ¬ B A)))
22 nntri3or 6011 . 2 ((A 𝜔 B 𝜔) → (A B A = B B A))
239, 15, 21, 22mpjao3dan 1201 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   = wceq 1242   wcel 1390  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-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257
This theorem is referenced by:  frec2uzf1od  8873
  Copyright terms: Public domain W3C validator