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Theorem pitric 7433

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 7421	. . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 7421	. . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		nntri2 6579	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
4	1, 2, 3	syl2an 289	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5		ltpiord 7431	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ 𝐴 ∈ 𝐵))
6		ltpiord 7431	. . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <_N 𝐴 ↔ 𝐵 ∈ 𝐴))
8	7	orbi2d 791	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9	8	notbid 668	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (¬ (𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
10	4, 5, 9	3bitr4d 220	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 104 ↔ wb 105 ∨ wo 709 = wceq 1372 ∈ wcel 2175 class class class wbr 4043 ωcom 4637 Ncnpi 7384 <_N clti 7387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-v 2773 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-tr 4142 df-eprel 4335 df-iord 4412 df-on 4414 df-suc 4417 df-iom 4638 df-xp 4680 df-ni 7416 df-lti 7419

This theorem is referenced by: (None)