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Theorem mainpart 38347
Description: Partition with general 𝑅 also imply member partition. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
Assertion
Ref Expression
mainpart (𝑅 Part 𝐴 → MembPart 𝐴)

Proof of Theorem mainpart
StepHypRef Expression
1 partimcomember 38339 . 2 (𝑅 Part 𝐴 → CoMembEr 𝐴)
2 mpet 38343 . 2 ( MembPart 𝐴 ↔ CoMembEr 𝐴)
31, 2sylibr 233 1 (𝑅 Part 𝐴 → MembPart 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   CoMembEr wcomember 37709   Part wpart 37720   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: (None)
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