Theorem fences2 37710

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 37701) 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 37709	. 2 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)
2		dfmembpart2 37635	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3	1, 2	sylib 217	1 ⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106  ∅c0 4322   ErALTV werALTV 37064   ElDisj weldisj 37074   MembPart wmembpart 37079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704  df-qs 8708  df-coss 37276  df-coels 37277  df-refrel 37377  df-cnvrefrel 37392  df-symrel 37409  df-trrel 37439  df-eqvrel 37450  df-coeleqvrel 37452  df-dmqs 37504  df-erALTV 37529  df-comember 37531  df-funALTV 37547  df-disjALTV 37570  df-eldisj 37572  df-part 37631  df-membpart 37633

This theorem is referenced by:  mainer2  37711