Theorem cvrcmp 37224

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 1189	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Poset)
2		simpl23 1251	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
3		simpl21 1249	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		simpl3l 1226	. . . . 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 37222	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≠ 𝑋)
8	1, 2, 3, 4, 7	syl31anc 1371	. . . 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 37219	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≤ 𝑋)
11	1, 2, 3, 4, 10	syl31anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≤ 𝑋)
12		simpr 484	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
13		simpl22 1250	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
14		simpl3r 1227	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑌)
15	5, 9, 6	cvrnbtwn4 37220	. . . . . . 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 1381	. . . . . 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 708	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 = 𝑋 ∨ 𝑋 = 𝑌))
18		neor 3035	. . . . 5 ⊢ ((𝑍 = 𝑋 ∨ 𝑋 = 𝑌) ↔ (𝑍 ≠ 𝑋 → 𝑋 = 𝑌))
19	17, 18	sylib 217	. . . 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 412	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
22		simp1 1134	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23		simp21 1204	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ∈ 𝐵)
24	5, 9	posref 17951	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
25	22, 23, 24	syl2anc 583	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ≤ 𝑋)
26		breq2 5074	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
27	25, 26	syl5ibcom 244	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
28	21, 27	impbid 211	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  Basecbs 16840  lecple 16895  Posetcpo 17940   ⋖ ccvr 37203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-proset 17928  df-poset 17946  df-plt 17963  df-covers 37207

This theorem is referenced by:  cvrcmp2  37225  atcmp  37252  llncmp  37463  lplncmp  37503  lvolcmp  37558  lhp2lt  37942