Theorem mainpart 39127

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 39119	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 39123	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 234	1 ⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   CoMembEr wcomember 38383   Part wpart 38394   MembPart wmembpart 38396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-qs 8641  df-coss 38671  df-coels 38672  df-refrel 38762  df-cnvrefrel 38777  df-symrel 38794  df-trrel 38828  df-eqvrel 38839  df-coeleqvrel 38841  df-dmqs 38893  df-erALTV 38919  df-comember 38921  df-funALTV 38937  df-disjALTV 38960  df-eldisj 38962  df-part 39039  df-membpart 39041

This theorem is referenced by: (None)