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Theorem pitric 7270

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 7258	. . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 7258	. . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		nntri2 6470	. . 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 7268	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_N 𝐵 ↔ 𝐴 ∈ 𝐵))
6		ltpiord 7268	. . . . 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 785	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 = 𝐵 ∨ 𝐵 <_N 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9	8	notbid 662	. 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 703 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ωcom 4572 Ncnpi 7221 <_N clti 7224

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-tr 4086 df-eprel 4272 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-ni 7253 df-lti 7256

This theorem is referenced by: (None)