![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opltcon1b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chpsscon1 30752 analog.) (Contributed by NM, 5-Nov-2011.) |
Ref | Expression |
---|---|
opltcon3.b | ⊢ 𝐵 = (Base‘𝐾) |
opltcon3.s | ⊢ < = (lt‘𝐾) |
opltcon3.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opltcon1b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opltcon3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opltcon3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opoccl 38059 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant3 1132 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | opltcon3.s | . . . 4 ⊢ < = (lt‘𝐾) | |
6 | 1, 5, 2 | opltcon3b 38069 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)))) |
7 | 4, 6 | syld3an2 1411 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)))) |
8 | 1, 2 | opococ 38060 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 8 | 3adant3 1132 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
10 | 9 | breq2d 5160 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)) ↔ ( ⊥ ‘𝑌) < 𝑋)) |
11 | 7, 10 | bitrd 278 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6543 Basecbs 17143 occoc 17204 ltcplt 18260 OPcops 38037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-proset 18247 df-poset 18265 df-plt 18282 df-oposet 38041 |
This theorem is referenced by: cvrcon3b 38142 |
Copyright terms: Public domain | W3C validator |