![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > polcon2N | Structured version Visualization version GIF version |
Description: Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
2polss.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polcon2N | ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2polss.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | 2polss.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | 1, 2 | 2polssN 39276 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → 𝑌 ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
4 | 3 | 3adant3 1129 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
5 | 1, 2 | polssatN 39269 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → ( ⊥ ‘𝑌) ⊆ 𝐴) |
6 | 5 | 3adant3 1129 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘𝑌) ⊆ 𝐴) |
7 | 1, 2 | polcon3N 39278 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑌) ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋)) |
8 | 6, 7 | syld3an2 1408 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → ( ⊥ ‘( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋)) |
9 | 4, 8 | sstrd 3984 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 ‘cfv 6533 Atomscatm 38623 HLchlt 38710 ⊥𝑃cpolN 39263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-psubsp 38864 df-pmap 38865 df-polarityN 39264 |
This theorem is referenced by: polcon2bN 39281 osumcllem3N 39319 |
Copyright terms: Public domain | W3C validator |