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Mirrors > Home > MPE Home > Th. List > 0pos | Structured version Visualization version GIF version |
Description: Technical lemma to simplify the statement of ipopos 17514. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16275) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
0pos | ⊢ ∅ ∈ Poset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5015 | . 2 ⊢ ∅ ∈ V | |
2 | ral0 4299 | . 2 ⊢ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎∅𝑎 ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑎) → 𝑎 = 𝑏) ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑐) → 𝑎∅𝑐)) | |
3 | base0 16276 | . . 3 ⊢ ∅ = (Base‘∅) | |
4 | df-ple 16326 | . . . 4 ⊢ le = Slot ;10 | |
5 | 4 | str0 16275 | . . 3 ⊢ ∅ = (le‘∅) |
6 | 3, 5 | ispos 17301 | . 2 ⊢ (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎∅𝑎 ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑎) → 𝑎 = 𝑏) ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑐) → 𝑎∅𝑐)))) |
7 | 1, 2, 6 | mpbir2an 704 | 1 ⊢ ∅ ∈ Poset |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 ∈ wcel 2166 ∀wral 3118 Vcvv 3415 ∅c0 4145 class class class wbr 4874 0cc0 10253 1c1 10254 ;cdc 11822 lecple 16313 Posetcpo 17294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-slot 16227 df-base 16229 df-ple 16326 df-poset 17300 |
This theorem is referenced by: ipopos 17514 |
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