Theorem oplecon1b 40082

Description: Contraposition law for strict ordering in orthoposets. (chsscon1 31990 analog.) (Contributed by NM, 6-Nov-2011.)

Hypotheses

Ref	Expression
opcon3.b	⊢ 𝐵 = (Base‘𝐾)
opcon3.l	⊢ ≤ = (le‘𝐾)
opcon3.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
oplecon1b	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋))

Proof of Theorem oplecon1b

Step	Hyp	Ref	Expression
1		opcon3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		opcon3.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
3	1, 2	opoccl 40075	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
4	3	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
5		opcon3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6	1, 5, 2	oplecon3b 40081	. . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋))))
7	4, 6	syld3an2 1438	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋))))
8	1, 2	opococ 40076	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
9	8	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
10	9	breq2d 5119	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)) ↔ ( ⊥ ‘𝑌) ≤ 𝑋))
11	7, 10	bitrd 282	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5107  ‘cfv 6537  Basecbs 17307  lecple 17355  occoc 17356  OPcops 40053

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-ext 2734  ax-nul 5267

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-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420  df-oposet 40057

This theorem is used by:  opoc1  40083  oldmm1  40098