Theorem mainpart 39420

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 39412	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 39416	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 236	1 ⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   CoMembEr wcomember 38676   Part wpart 38687   MembPart wmembpart 38689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-eprel 5545  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ec 8675  df-qs 8679  df-coss 38964  df-coels 38965  df-refrel 39055  df-cnvrefrel 39070  df-symrel 39087  df-trrel 39121  df-eqvrel 39132  df-coeleqvrel 39134  df-dmqs 39186  df-erALTV 39212  df-comember 39214  df-funALTV 39230  df-disjALTV 39253  df-eldisj 39255  df-part 39332  df-membpart 39334

This theorem is referenced by: (None)