Theorem mainpart 40155

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 40147	. 2 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)
2		mpet 40151	. 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 39411   Part wpart 39422   MembPart wmembpart 39424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761  df-coss 39699  df-coels 39700  df-refrel 39790  df-cnvrefrel 39805  df-symrel 39822  df-trrel 39856  df-eqvrel 39867  df-coeleqvrel 39869  df-dmqs 39921  df-erALTV 39947  df-comember 39949  df-funALTV 39965  df-disjALTV 39988  df-eldisj 39990  df-part 40067  df-membpart 40069

This theorem is used by: (None)