Theorem cvrcmp 39782

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 1198	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Poset)
2		simpl23 1260	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
3		simpl21 1258	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		simpl3l 1235	. . . . 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 39780	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≠ 𝑋)
8	1, 2, 3, 4, 7	syl31anc 1381	. . . 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 39777	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≤ 𝑋)
11	1, 2, 3, 4, 10	syl31anc 1381	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≤ 𝑋)
12		simpr 485	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
13		simpl22 1259	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
14		simpl3r 1236	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑌)
15	5, 9, 6	cvrnbtwn4 39778	. . . . . . 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 1391	. . . . . 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 718	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 = 𝑋 ∨ 𝑋 = 𝑌))
18		neor 3027	. . . . 5 ⊢ ((𝑍 = 𝑋 ∨ 𝑋 = 𝑌) ↔ (𝑍 ≠ 𝑋 → 𝑋 = 𝑌))
19	17, 18	sylib 219	. . . 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 413	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
22		simp1 1142	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23		simp21 1213	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ∈ 𝐵)
24	5, 9	posref 18282	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
25	22, 23, 24	syl2anc 590	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ≤ 𝑋)
26		breq2 5083	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
27	25, 26	syl5ibcom 246	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
28	21, 27	impbid 213	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   class class class wbr 5079  ‘cfv 6492  Basecbs 17177  lecple 17225  Posetcpo 18271   ⋖ ccvr 39761

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-proset 18258  df-poset 18277  df-plt 18292  df-covers 39765

This theorem is referenced by:  cvrcmp2  39783  atcmp  39810  llncmp  40021  lplncmp  40061  lvolcmp  40116  lhp2lt  40500