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Mirrors > Home > MPE Home > Th. List > Mathboxes > fences2 | Structured version Visualization version GIF version |
Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 38340) generate a partition of the members, it alo means that (𝑅 ErALTV 𝐴 → ElDisj 𝐴) and that (𝑅 ErALTV 𝐴 → ¬ ∅ ∈ 𝐴). (Contributed by Peter Mazsa, 15-Oct-2021.) |
Ref | Expression |
---|---|
fences2 | ⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fences 38348 | . 2 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴) | |
2 | dfmembpart2 38274 | . 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
3 | 1, 2 | sylib 217 | 1 ⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2098 ∅c0 4326 ErALTV werALTV 37707 ElDisj weldisj 37717 MembPart wmembpart 37722 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-eprel 5586 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8733 df-qs 8737 df-coss 37915 df-coels 37916 df-refrel 38016 df-cnvrefrel 38031 df-symrel 38048 df-trrel 38078 df-eqvrel 38089 df-coeleqvrel 38091 df-dmqs 38143 df-erALTV 38168 df-comember 38170 df-funALTV 38186 df-disjALTV 38209 df-eldisj 38211 df-part 38270 df-membpart 38272 |
This theorem is referenced by: mainer2 38350 |
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