Theorem fences2 38841

Description: The Theorem of Fences by Equivalences: all conceivable equivalence relations (besides the comember equivalence relation cf. mpet3 38832) 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 38840	. 2 ⊢ (𝑅 ErALTV 𝐴 → MembPart 𝐴)
2		dfmembpart2 38766	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3	1, 2	sylib 218	1 ⊢ (𝑅 ErALTV 𝐴 → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108  ∅c0 4342   ErALTV werALTV 38202   ElDisj weldisj 38212   MembPart wmembpart 38217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-eprel 5593  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-ec 8755  df-qs 8759  df-coss 38407  df-coels 38408  df-refrel 38508  df-cnvrefrel 38523  df-symrel 38540  df-trrel 38570  df-eqvrel 38581  df-coeleqvrel 38583  df-dmqs 38635  df-erALTV 38660  df-comember 38662  df-funALTV 38678  df-disjALTV 38701  df-eldisj 38703  df-part 38762  df-membpart 38764

This theorem is referenced by:  mainer2  38842