nntri2 - Intuitionistic Logic Explorer

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Theorem nntri2 5988

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 4208	. . . . 5 ⊢ ¬ A ∈ A
2		eleq2 2083	. . . . 5 ⊢ (A = B → (A ∈ A ↔ A ∈ B))
3	1, 2	mtbii 586	. . . 4 ⊢ (A = B → ¬ A ∈ B)
4	3	con2i 545	. . 3 ⊢ (A ∈ B → ¬ A = B)
5		en2lp 4216	. . . 4 ⊢ ¬ (A ∈ B ∧ B ∈ A)
6	5	imnani 612	. . 3 ⊢ (A ∈ B → ¬ B ∈ A)
7		ioran 656	. . 3 ⊢ (¬ (A = B ∨ B ∈ A) ↔ (¬ A = B ∧ ¬ B ∈ A))
8	4, 6, 7	sylanbrc 396	. 2 ⊢ (A ∈ B → ¬ (A = B ∨ B ∈ A))
9		nntri3or 5987	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ∈ B ∨ A = B ∨ B ∈ A))
10		3orass 876	. . . . 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 635	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A = B ∨ B ∈ A) ∨ A ∈ B))
13	12	ord 630	. 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 616 ∨ w3o 872 = wceq 1228 ∈ wcel 1374 𝜔com 4240

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 532 ax-in2 533 ax-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1363 ax-ie2 1364 ax-8 1376 ax-10 1377 ax-11 1378 ax-i12 1379 ax-bnd 1380 ax-4 1381 ax-13 1385 ax-14 1386 ax-17 1400 ax-i9 1404 ax-ial 1409 ax-i5r 1410 ax-ext 2004 ax-sep 3849 ax-nul 3857 ax-pow 3901 ax-pr 3918 ax-un 4120 ax-setind 4204 ax-iinf 4238

This theorem depends on definitions: df-bi 110 df-3or 874 df-3an 875 df-tru 1231 df-nf 1330 df-sb 1628 df-clab 2009 df-cleq 2015 df-clel 2018 df-nfc 2149 df-ne 2188 df-ral 2289 df-rex 2290 df-v 2537 df-dif 2897 df-un 2899 df-in 2901 df-ss 2908 df-nul 3202 df-pw 3336 df-sn 3356 df-pr 3357 df-uni 3555 df-int 3590 df-tr 3829 df-iord 4052 df-on 4054 df-suc 4057 df-iom 4241

This theorem is referenced by: nnaord 5993 nnmord 6001 pitric 6181