sotri2 - Intuitionistic Logic Explorer

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Theorem sotri2 5001

Description: A transitivity relation. (Read ¬ B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

**Hypotheses**
Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

**Assertion**
Ref	Expression
sotri2	⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

**Proof of Theorem sotri2**
Step	Hyp	Ref	Expression
1		simp2 988	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐴)
2		soi.2	. . . . . . 7 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
3	2	brel 4656	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	3ad2ant3 1010	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5		simp1 987	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴 ∈ 𝑆)
6		df-3an 970	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ↔ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ 𝐴 ∈ 𝑆))
7	4, 5, 6	sylanbrc 414	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
8		simp3 989	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
9		soi.1	. . . . 5 ⊢ 𝑅 Or 𝑆
10		sowlin 4298	. . . . 5 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶)))
11	9, 10	mpan 421	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶)))
12	7, 8, 11	sylc 62	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶))
13	12	ord 714	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶))
14	1, 13	mpd 13	1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 968 ∈ wcel 2136 ⊆ wss 3116 class class class wbr 3982 Or wor 4273 × cxp 4602

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-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-br 3983 df-opab 4044 df-iso 4275 df-xp 4610

This theorem is referenced by: (None)