sotri2 - Intuitionistic Logic Explorer

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Theorem sotri2 5134

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 1024	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐴)
2		soi.2	. . . . . . 7 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
3	2	brel 4778	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
4	3	3ad2ant3 1046	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
5		simp1 1023	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴 ∈ 𝑆)
6		df-3an 1006	. . . . 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 1025	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
9		soi.1	. . . . 5 ⊢ 𝑅 Or 𝑆
10		sowlin 4417	. . . . 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 731	. 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 715 ∧ w3a 1004 ∈ wcel 2202 ⊆ wss 3200 class class class wbr 4088 Or wor 4392 × cxp 4723

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 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-br 4089 df-opab 4151 df-iso 4394 df-xp 4731

This theorem is referenced by: (None)