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Mirrors > Home > MPE Home > Th. List > psref | Structured version Visualization version GIF version |
Description: A poset is reflexive. (Contributed by NM, 13-May-2008.) |
Ref | Expression |
---|---|
psref.1 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
psref | ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psref.1 | . . . . 5 ⊢ 𝑋 = dom 𝑅 | |
2 | psdmrn 18631 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) | |
3 | 2 | simpld 494 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
4 | 1, 3 | eqtrid 2787 | . . . 4 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅) |
5 | 4 | eleq2d 2825 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅)) |
6 | pslem 18630 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴 = 𝐴))) | |
7 | 6 | simp2d 1142 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴)) |
8 | 5, 7 | sylbid 240 | . 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 → 𝐴𝑅𝐴)) |
9 | 8 | imp 406 | 1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 class class class wbr 5148 dom cdm 5689 ran crn 5690 PosetRelcps 18622 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ps 18624 |
This theorem is referenced by: psss 18638 psssdm2 18639 ordtt1 23403 |
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