Theorem fences2 38227

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 38218) 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 38226	. 2 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)
2		dfmembpart2 38152	. 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 395   ∈ wcel 2098  ∅c0 4317   ErALTV werALTV 37581   ElDisj weldisj 37591   MembPart wmembpart 37596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704  df-qs 8708  df-coss 37793  df-coels 37794  df-refrel 37894  df-cnvrefrel 37909  df-symrel 37926  df-trrel 37956  df-eqvrel 37967  df-coeleqvrel 37969  df-dmqs 38021  df-erALTV 38046  df-comember 38048  df-funALTV 38064  df-disjALTV 38087  df-eldisj 38089  df-part 38148  df-membpart 38150

This theorem is referenced by:  mainer2  38228