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Theorem isposi 18503

Description: Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)

**Hypotheses**
Ref	Expression
isposi.k	⊢ 𝐾 ∈ V
isposi.b	⊢ 𝐵 = (Base‘𝐾)
isposi.l	⊢ ≤ = (le‘𝐾)
isposi.1	⊢ (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥)
isposi.2	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
isposi.3	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))

**Assertion**
Ref	Expression
isposi	⊢ 𝐾 ∈ Poset

Distinct variable groups: 𝑥,𝑦,𝑧,𝐵 𝑥, ≤ ,𝑦,𝑧 Allowed substitution hints: 𝐾(𝑥, 𝑦, 𝑧)

**Proof of Theorem isposi**
Step	Hyp	Ref	Expression
1		isposi.k	. 2 ⊢ 𝐾 ∈ V
2		isposi.1	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥)
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑥 ≤ 𝑥)
4		isposi.2	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
5	4	3adant3 1150	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
6		isposi.3	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
7	3, 5, 6	3jca 1146	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
8	7	rgen3 3207	. 2 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
9		isposi.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
10		isposi.l	. . 3 ⊢ ≤ = (le‘𝐾)
11	9, 10	ispos 18494	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
12	1, 8, 11	mpbir2an 724	1 ⊢ 𝐾 ∈ Poset

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 class class class wbr 5102 ‘cfv 6539 Basecbs 17393 lecple 17441 Posetcpo 18487

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-nul 5259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-iota 6494 df-fv 6547 df-poset 18493

This theorem is used by: isposix 18504 xrge0omnd 21757 xrstos 33618

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