Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 0pos | Structured version Visualization version GIF version |
Description: Technical lemma to simplify the statement of ipopos 17764. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16529) 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 5204 | . 2 ⊢ ∅ ∈ V | |
2 | ral0 4456 | . 2 ⊢ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎∅𝑎 ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑎) → 𝑎 = 𝑏) ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑐) → 𝑎∅𝑐)) | |
3 | base0 16530 | . . 3 ⊢ ∅ = (Base‘∅) | |
4 | df-ple 16579 | . . . 4 ⊢ le = Slot ;10 | |
5 | 4 | str0 16529 | . . 3 ⊢ ∅ = (le‘∅) |
6 | 3, 5 | ispos 17551 | . 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 709 | 1 ⊢ ∅ ∈ Poset |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∅c0 4291 class class class wbr 5059 0cc0 10531 1c1 10532 ;cdc 12092 lecple 16566 Posetcpo 17544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-slot 16481 df-base 16483 df-ple 16579 df-poset 17550 |
This theorem is referenced by: ipopos 17764 |
Copyright terms: Public domain | W3C validator |