Theorem fences2 39636

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 39627) generate a partition of the members, it alo means that (𝑅 ErALTV 𝐴 → ElDisj 𝐴) and that (𝑅 ErALTV 𝐴 → ¬ ∅ ∈ 𝐴). (Contributed by Peter Mazsa, 15-Oct-2021.)

Assertion

Ref	Expression
fences2	⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Proof of Theorem fences2

Step	Hyp	Ref	Expression
1		fences 39635	. 2 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)
2		dfmembpart2 39550	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3	1, 2	sylib 221	1 ⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2142  ∅c0 4285   ErALTV werALTV 38886   ElDisj weldisj 38898   MembPart wmembpart 38903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  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 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ec 8694  df-qs 8698  df-coss 39178  df-coels 39179  df-refrel 39269  df-cnvrefrel 39284  df-symrel 39301  df-trrel 39335  df-eqvrel 39346  df-coeleqvrel 39348  df-dmqs 39400  df-erALTV 39426  df-comember 39428  df-funALTV 39444  df-disjALTV 39467  df-eldisj 39469  df-part 39546  df-membpart 39548

This theorem is used by:  mainer2  39637