Theorem cvrcmp 37034

Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)

Hypotheses

Ref	Expression
cvrcmp.b	⊢ 𝐵 = (Base‘𝐾)
cvrcmp.l	⊢ ≤ = (le‘𝐾)
cvrcmp.c	⊢ 𝐶 = ( ⋖ ‘𝐾)

Assertion

Ref	Expression
cvrcmp	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Proof of Theorem cvrcmp

Step	Hyp	Ref	Expression
1		simpl1 1193	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Poset)
2		simpl23 1255	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
3		simpl21 1253	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		simpl3l 1230	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑋)
5		cvrcmp.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		cvrcmp.c	. . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾)
7	5, 6	cvrne 37032	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≠ 𝑋)
8	1, 2, 3, 4, 7	syl31anc 1375	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≠ 𝑋)
9		cvrcmp.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	5, 9, 6	cvrle 37029	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≤ 𝑋)
11	1, 2, 3, 4, 10	syl31anc 1375	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≤ 𝑋)
12		simpr 488	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
13		simpl22 1254	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
14		simpl3r 1231	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑌)
15	5, 9, 6	cvrnbtwn4 37030	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) ↔ (𝑍 = 𝑋 ∨ 𝑋 = 𝑌)))
16	1, 2, 13, 3, 14, 15	syl131anc 1385	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → ((𝑍 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) ↔ (𝑍 = 𝑋 ∨ 𝑋 = 𝑌)))
17	11, 12, 16	mpbi2and 712	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 = 𝑋 ∨ 𝑋 = 𝑌))
18		neor 3033	. . . . 5 ⊢ ((𝑍 = 𝑋 ∨ 𝑋 = 𝑌) ↔ (𝑍 ≠ 𝑋 → 𝑋 = 𝑌))
19	17, 18	sylib 221	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 ≠ 𝑋 → 𝑋 = 𝑌))
20	8, 19	mpd 15	. . 3 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 = 𝑌)
21	20	ex 416	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
22		simp1 1138	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23		simp21 1208	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ∈ 𝐵)
24	5, 9	posref 17825	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
25	22, 23, 24	syl2anc 587	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ≤ 𝑋)
26		breq2 5057	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
27	25, 26	syl5ibcom 248	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
28	21, 27	impbid 215	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940   class class class wbr 5053  ‘cfv 6380  Basecbs 16760  lecple 16809  Posetcpo 17814   ⋖ ccvr 37013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-proset 17802  df-poset 17820  df-plt 17836  df-covers 37017

This theorem is referenced by:  cvrcmp2  37035  atcmp  37062  llncmp  37273  lplncmp  37313  lvolcmp  37368  lhp2lt  37752