Theorem fences 38316

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet 38311) 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 38306	. 2 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)
2		mpet 38311	. 2 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)
3	1, 2	sylibr 233	1 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)

Syntax hints:   → wi 4   ErALTV werALTV 37674   CoMembEr wcomember 37676   MembPart wmembpart 37689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731  df-coss 37883  df-coels 37884  df-refrel 37984  df-cnvrefrel 37999  df-symrel 38016  df-trrel 38046  df-eqvrel 38057  df-coeleqvrel 38059  df-dmqs 38111  df-erALTV 38136  df-comember 38138  df-funALTV 38154  df-disjALTV 38177  df-eldisj 38179  df-part 38238  df-membpart 38240

This theorem is referenced by:  fences2  38317