Theorem fences 39322

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 39317) 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 39312	. 2 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
2		mpet 39317	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 235	1 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   ErALTV werALTV 38573   CoMembEr wcomember 38577   MembPart wmembpart 38590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-ral 3051  df-rex 3061  df-rmo 3341  df-rab 3389  df-v 3430  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8638  df-qs 8642  df-coss 38865  df-coels 38866  df-refrel 38956  df-cnvrefrel 38971  df-symrel 38988  df-trrel 39022  df-eqvrel 39033  df-coeleqvrel 39035  df-dmqs 39087  df-erALTV 39113  df-comember 39115  df-funALTV 39131  df-disjALTV 39154  df-eldisj 39156  df-part 39233  df-membpart 39235

This theorem is referenced by:  fences2  39323