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Theorem pitric 6977

Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)

**Assertion**
Ref	Expression
pitric	⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴)))

**Proof of Theorem pitric**
Step	Hyp	Ref	Expression
1		pinn 6965	. . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 6965	. . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		nntri2 6295	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
4	1, 2, 3	syl2an 284	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5		ltpiord 6975	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ 𝐴 ∈ 𝐵))
6		ltpiord 6975	. . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	ancoms 265	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
8	7	orbi2d 742	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9	8	notbid 630	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (¬ (𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
10	4, 5, 9	3bitr4d 219	1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 ωcom 4433 Ncnpi 6928 <_N clti 6931

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431

This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-tr 3959 df-eprel 4140 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-ni 6960 df-lti 6963

This theorem is referenced by: (None)