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Theorem isposi 18233

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 1133	. . . 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 1132	. . . 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 1128	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
8	7	rgen3 3178	. 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 18224	. 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 711	1 ⊢ 𝐾 ∈ Poset

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3048 Vcvv 3437 class class class wbr 5095 ‘cfv 6488 Basecbs 17124 lecple 17172 Posetcpo 18217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2705 ax-nul 5248

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2712 df-cleq 2725 df-clel 2808 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-iota 6444 df-fv 6496 df-poset 18223

This theorem is referenced by: isposix 18234 xrge0omnd 21386 xrstos 33000

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