Theorem dfmember 36711

Description: Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
dfmember	⊢ ( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Proof of Theorem dfmember

Step	Hyp	Ref	Expression
1		df-member 36705	. 2 ⊢ ( MembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
2		df-coels 36465	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	erALTVeq1i 36709	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 3	bitr4i 277	1 ⊢ ( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Syntax hints:   ↔ wb 205   E cep 5485  ^◡ccnv 5579   ↾ cres 5582   ≀ ccoss 36260   ∼ ccoels 36261   ErALTV werALTV 36286   MembEr wmember 36288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458  df-qs 8462  df-coels 36465  df-refrel 36557  df-symrel 36585  df-trrel 36615  df-eqvrel 36625  df-dmqs 36679  df-erALTV 36703  df-member 36705

This theorem is referenced by:  dfmember2  36712