sotri3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sotri3		Structured version Visualization version GIF version

Theorem sotri3 6032

Description: A transitivity relation. (Read 𝐴 < 𝐵 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
sotri3	⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

**Proof of Theorem sotri3**
Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5651	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	simprd 495	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 5530	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
6	4, 5	mpan 686	. . . . . 6 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
7	6	con2bid 354	. . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8		breq2 5082	. . . . . . 7 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐴𝑅𝐵))
9	8	biimprd 247	. . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
10	4, 1	sotri 6029	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	expcom 413	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
12	9, 11	jaoi 853	. . . . 5 ⊢ ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
13	7, 12	syl6bir 253	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)))
14	13	com3r 87	. . 3 ⊢ (𝐴𝑅𝐵 → ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
15	3, 14	mpan2d 690	. 2 ⊢ (𝐴𝑅𝐵 → (𝐶 ∈ 𝑆 → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
16	15	3imp21 1112	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ⊆ wss 3891 class class class wbr 5078 Or wor 5501 × cxp 5586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-po 5502 df-so 5503 df-xp 5594

This theorem is referenced by: archnq 10720

W3C validator