Theorem mainpart 37082

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

Step	Hyp	Ref	Expression
1		partimcomember 37074	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 37078	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 233	1 ⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   CoMembEr wcomember 36438   Part wpart 36449   MembPart wmembpart 36451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-id 5506  df-eprel 5512  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ec 8549  df-qs 8553  df-coss 36650  df-coels 36651  df-refrel 36751  df-cnvrefrel 36766  df-symrel 36783  df-trrel 36813  df-eqvrel 36824  df-coeleqvrel 36826  df-dmqs 36878  df-erALTV 36903  df-comember 36905  df-funALTV 36921  df-disjALTV 36944  df-eldisj 36946  df-part 37005  df-membpart 37007

This theorem is referenced by: (None)