Theorem cvrcmp 40156

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 1210	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Poset)
2		simpl23 1272	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
3		simpl21 1270	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		simpl3l 1247	. . . . 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 40154	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≠ 𝑋)
8	1, 2, 3, 4, 7	syl31anc 1400	. . . 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 40151	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≤ 𝑋)
11	1, 2, 3, 4, 10	syl31anc 1400	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≤ 𝑋)
12		simpr 490	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
13		simpl22 1271	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
14		simpl3r 1248	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑌)
15	5, 9, 6	cvrnbtwn4 40152	. . . . . . 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 1410	. . . . . 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 725	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 = 𝑋 ∨ 𝑋 = 𝑌))
18		neor 3047	. . . . 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 16	. . 3 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 = 𝑌)
21	20	ex 418	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
22		simp1 1154	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23		simp21 1225	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ∈ 𝐵)
24	5, 9	posref 18406	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
25	22, 23, 24	syl2anc 596	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ≤ 𝑋)
26		breq2 5107	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ‘cfv 6533  Basecbs 17301  lecple 17349  Posetcpo 18395   ⋖ ccvr 40135

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-proset 18382  df-poset 18401  df-plt 18416  df-covers 40139

This theorem is used by:  cvrcmp2  40157  atcmp  40184  llncmp  40395  lplncmp  40435  lvolcmp  40490  lhp2lt  40874