Theorem mainpart 38782

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 38774	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 38778	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 234	1 ⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   CoMembEr wcomember 38148   Part wpart 38159   MembPart wmembpart 38161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-id 5545  df-eprel 5550  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8715  df-qs 8719  df-coss 38350  df-coels 38351  df-refrel 38451  df-cnvrefrel 38466  df-symrel 38483  df-trrel 38513  df-eqvrel 38524  df-coeleqvrel 38526  df-dmqs 38578  df-erALTV 38603  df-comember 38605  df-funALTV 38621  df-disjALTV 38644  df-eldisj 38646  df-part 38705  df-membpart 38707

This theorem is referenced by: (None)