sotri2 - Intuitionistic Logic Explorer

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Theorem sotri2 5099

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 1001	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐴)
2		soi.2	. . . . . . 7 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
3	2	brel 4745	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	3ad2ant3 1023	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5		simp1 1000	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴 ∈ 𝑆)
6		df-3an 983	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ↔ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ 𝐴 ∈ 𝑆))
7	4, 5, 6	sylanbrc 417	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
8		simp3 1002	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
9		soi.1	. . . . 5 ⊢ 𝑅 Or 𝑆
10		sowlin 4385	. . . . 5 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶)))
11	9, 10	mpan 424	. . . 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 726	. 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 104 ∨ wo 710 ∧ w3a 981 ∈ wcel 2178 ⊆ wss 3174 class class class wbr 4059 Or wor 4360 × cxp 4691

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-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-br 4060 df-opab 4122 df-iso 4362 df-xp 4699

This theorem is referenced by: (None)