Theorem fences 38808

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 38803) 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 38798	. 2 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
2		mpet 38803	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 234	1 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   ErALTV werALTV 38171   CoMembEr wcomember 38173   MembPart wmembpart 38186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-eprel 5553  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8719  df-qs 8723  df-coss 38375  df-coels 38376  df-refrel 38476  df-cnvrefrel 38491  df-symrel 38508  df-trrel 38538  df-eqvrel 38549  df-coeleqvrel 38551  df-dmqs 38603  df-erALTV 38628  df-comember 38630  df-funALTV 38646  df-disjALTV 38669  df-eldisj 38671  df-part 38730  df-membpart 38732

This theorem is referenced by:  fences2  38809