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Mirrors > Home > MPE Home > Th. List > Mathboxes > opltcon1b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chpsscon1 29208 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 36210 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant3 1124 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | opltcon3.s | . . . 4 ⊢ < = (lt‘𝐾) | |
6 | 1, 5, 2 | opltcon3b 36220 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)))) |
7 | 4, 6 | syld3an2 1403 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)))) |
8 | 1, 2 | opococ 36211 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 8 | 3adant3 1124 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
10 | 9 | breq2d 5069 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) < ( ⊥ ‘( ⊥ ‘𝑋)) ↔ ( ⊥ ‘𝑌) < 𝑋)) |
11 | 7, 10 | bitrd 280 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 occoc 16561 ltcplt 17539 OPcops 36188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-proset 17526 df-poset 17544 df-plt 17556 df-oposet 36192 |
This theorem is referenced by: cvrcon3b 36293 |
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