Theorem fences 37062

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 37057) 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 37052	. 2 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
2		mpet 37057	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 233	1 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   ErALTV werALTV 36415   CoMembEr wcomember 36417   MembPart wmembpart 36430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-id 5507  df-eprel 5513  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ec 8548  df-qs 8552  df-coss 36629  df-coels 36630  df-refrel 36730  df-cnvrefrel 36745  df-symrel 36762  df-trrel 36792  df-eqvrel 36803  df-coeleqvrel 36805  df-dmqs 36857  df-erALTV 36882  df-comember 36884  df-funALTV 36900  df-disjALTV 36923  df-eldisj 36925  df-part 36984  df-membpart 36986

This theorem is referenced by:  fences2  37063