Theorem mainpart 39634

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 39626	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 39630	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 237	1 ⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

This proof depends on syntax axioms:   → wi 4   CoMembEr wcomember 38890   Part wpart 38901   MembPart wmembpart 38903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-coss 39178  df-coels 39179  df-refrel 39269  df-cnvrefrel 39284  df-symrel 39301  df-trrel 39335  df-eqvrel 39346  df-coeleqvrel 39348  df-dmqs 39400  df-erALTV 39426  df-comember 39428  df-funALTV 39444  df-disjALTV 39467  df-eldisj 39469  df-part 39546  df-membpart 39548

This theorem is used by: (None)