nntri2 - Intuitionistic Logic Explorer

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Theorem nntri2 6012

Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)

**Assertion**
Ref	Expression
nntri2	⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))

**Proof of Theorem nntri2**
Step	Hyp	Ref	Expression
1		elirr 4224	. . . . 5 ⊢ ¬ A ∈ A
2		eleq2 2098	. . . . 5 ⊢ (A = B → (A ∈ A ↔ A ∈ B))
3	1, 2	mtbii 598	. . . 4 ⊢ (A = B → ¬ A ∈ B)
4	3	con2i 557	. . 3 ⊢ (A ∈ B → ¬ A = B)
5		en2lp 4232	. . . 4 ⊢ ¬ (A ∈ B ∧ B ∈ A)
6	5	imnani 624	. . 3 ⊢ (A ∈ B → ¬ B ∈ A)
7		ioran 668	. . 3 ⊢ (¬ (A = B ∨ B ∈ A) ↔ (¬ A = B ∧ ¬ B ∈ A))
8	4, 6, 7	sylanbrc 394	. 2 ⊢ (A ∈ B → ¬ (A = B ∨ B ∈ A))
9		nntri3or 6011	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ∨ A = B ∨ B ∈ A))
10		3orass 887	. . . . 5 ⊢ ((A ∈ B ∨ A = B ∨ B ∈ A) ↔ (A ∈ B ∨ (A = B ∨ B ∈ A)))
11	9, 10	sylib 127	. . . 4 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ∨ (A = B ∨ B ∈ A)))
12	11	orcomd 647	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A = B ∨ B ∈ A) ∨ A ∈ B))
13	12	ord 642	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (¬ (A = B ∨ B ∈ A) → A ∈ B))
14	8, 13	impbid2 131	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 ∨ wo 628 ∨ w3o 883 = 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: nnaord 6018 nnmord 6026 pitric 6305