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Mirrors > Home > MPE Home > Th. List > Mathboxes > oplecon1b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 29281 analog.) (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
opcon3.b | ⊢ 𝐵 = (Base‘𝐾) |
opcon3.l | ⊢ ≤ = (le‘𝐾) |
opcon3.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
oplecon1b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opcon3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opcon3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opoccl 36334 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | opcon3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | 1, 5, 2 | oplecon3b 36340 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
7 | 4, 6 | syld3an2 1407 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
8 | 1, 2 | opococ 36335 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 8 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
10 | 9 | breq2d 5081 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)) ↔ ( ⊥ ‘𝑌) ≤ 𝑋)) |
11 | 7, 10 | bitrd 281 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 Basecbs 16486 lecple 16575 occoc 16576 OPcops 36312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 df-oposet 36316 |
This theorem is referenced by: opoc1 36342 oldmm1 36357 |
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