Theorem oplecon1b 34488

Description: Contraposition law for strict ordering in orthoposets. (chsscon1 28360 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 34481	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
4	3	3adant3 1081	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
5		opcon3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6	1, 5, 2	oplecon3b 34487	. . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋))))
7	4, 6	syld3an2 1373	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋))))
8	1, 2	opococ 34482	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
9	8	3adant3 1081	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
10	9	breq2d 4665	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)) ↔ ( ⊥ ‘𝑌) ≤ 𝑋))
11	7, 10	bitrd 268	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   class class class wbr 4653  ‘cfv 5888  Basecbs 15857  lecple 15948  occoc 15949  OPcops 34459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oposet 34463

This theorem is referenced by:  opoc1  34489  oldmm1  34504