Theorem fences 39575

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 39570) generate a partition of the members. (Contributed by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
fences	⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Proof of Theorem fences

Step	Hyp	Ref	Expression
1		mainer 39565	. 2 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
2		mpet 39570	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 237	1 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   ErALTV werALTV 38826   CoMembEr wcomember 38830   MembPart wmembpart 38843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  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 8695  df-qs 8699  df-coss 39118  df-coels 39119  df-refrel 39209  df-cnvrefrel 39224  df-symrel 39241  df-trrel 39275  df-eqvrel 39286  df-coeleqvrel 39288  df-dmqs 39340  df-erALTV 39366  df-comember 39368  df-funALTV 39384  df-disjALTV 39407  df-eldisj 39409  df-part 39486  df-membpart 39488

This theorem is referenced by:  fences2  39576