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Theorem pitric 7153

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 7141	. . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 7141	. . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		nntri2 6398	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
4	1, 2, 3	syl2an 287	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5		ltpiord 7151	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ 𝐴 ∈ 𝐵))
6		ltpiord 7151	. . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
8	7	orbi2d 780	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9	8	notbid 657	. 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 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ωcom 4512 Ncnpi 7104 <_N clti 7107

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-eprel 4219 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-ni 7136 df-lti 7139

This theorem is referenced by: (None)