Theorem cvrcmp 33588

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 1057	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ Poset)
2		simpl23 1134	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ∈ 𝐵)
3		simpl21 1132	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
4		simpl3l 1109	. . . . 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 33586	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≠ 𝑋)
8	1, 2, 3, 4, 7	syl31anc 1321	. . . 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 33583	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 ≤ 𝑋)
11	1, 2, 3, 4, 10	syl31anc 1321	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍 ≤ 𝑋)
12		simpr 476	. . . . . 6 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑌)
13		simpl22 1133	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
14		simpl3r 1110	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → 𝑍𝐶𝑌)
15	5, 9, 6	cvrnbtwn4 33584	. . . . . . 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 1331	. . . . . 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 958	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) ∧ 𝑋 ≤ 𝑌) → (𝑍 = 𝑋 ∨ 𝑋 = 𝑌))
18		neor 2873	. . . . 5 ⊢ ((𝑍 = 𝑋 ∨ 𝑋 = 𝑌) ↔ (𝑍 ≠ 𝑋 → 𝑋 = 𝑌))
19	17, 18	sylib 207	. . . 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 449	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
22		simp1 1054	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23		simp21 1087	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ∈ 𝐵)
24	5, 9	posref 16774	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
25	22, 23, 24	syl2anc 691	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → 𝑋 ≤ 𝑋)
26		breq2 4587	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
27	25, 26	syl5ibcom 234	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
28	21, 27	impbid 201	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763   ⋖ ccvr 33567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-covers 33571

This theorem is referenced by:  cvrcmp2  33589  atcmp  33616  llncmp  33826  lplncmp  33866  lvolcmp  33921  lhp2lt  34305