Theorem posi 17676

Description: Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)

Hypotheses

Ref	Expression
posi.b	⊢ 𝐵 = (Base‘𝐾)
posi.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
posi	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Proof of Theorem posi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		posi.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		posi.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3	1, 2	ispos 17673	. . 3 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
4	3	simprbi 500	. 2 ⊢ (𝐾 ∈ Poset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
5		breq1 5033	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
6		breq2 5034	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑋 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
7	5, 6	bitrd 282	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋))
8		breq1 5033	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		breq2 5034	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	8, 9	anbi12d 634	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ (𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋)))
11		eqeq1 2742	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
12	10, 11	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ↔ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋) → 𝑋 = 𝑦)))
13	8	anbi1d 633	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧)))
14		breq1 5033	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
15	13, 14	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
16	7, 12, 15	3anbi123d 1437	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋) → 𝑋 = 𝑦) ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
17		breq2 5034	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		breq1 5033	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
19	17, 18	anbi12d 634	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋)))
20		eqeq2 2750	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
21	19, 20	imbi12d 348	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋) → 𝑋 = 𝑦) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌)))
22		breq1 5033	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
23	17, 22	anbi12d 634	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧)))
24	23	imbi1d 345	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)))
25	21, 24	3anbi23d 1440	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑋) → 𝑋 = 𝑦) ∧ ((𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧))))
26		breq2 5034	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍))
27	26	anbi2d 632	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)))
28		breq2 5034	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
29	27, 28	imbi12d 348	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
30	29	3anbi3d 1443	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧) → 𝑋 ≤ 𝑧)) ↔ (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))))
31	16, 25, 30	rspc3v 3539	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))))
32	4, 31	mpan9 510	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  Vcvv 3398   class class class wbr 5030  ‘cfv 6339  Basecbs 16586  lecple 16675  Posetcpo 17666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-poset 17672

This theorem is referenced by:  posasymb  17678  postr  17679