Theorem epeli 5526

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5525. (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
epeli.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
epeli	⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Proof of Theorem epeli

Step	Hyp	Ref	Expression
1		epeli.1	. 2 ⊢ 𝐵 ∈ V
2		epelg 5525	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086   E cep 5523

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-eprel 5524

This theorem is referenced by:  epel  5527  0sn0ep  5528  smoiso  8295  smoiso2  8302  ecid  8720  ordiso2  9423  cantnflt  9584  cantnfp1lem3  9592  oemapso  9594  cantnflem1b  9598  cantnflem1  9601  cantnf  9605  wemapwe  9609  cnfcomlem  9611  cnfcom  9612  cnfcom3lem  9615  leweon  9924  r0weon  9925  alephiso  10011  fin23lem27  10241  fpwwe2lem8  10552  oniso  28277  ex-eprel  30518  cardpred  35251  satefvfmla0  35616  satefvfmla1  35623  dftr6  35949  coep  35950  coepr  35951  brsset  36085  brtxpsd  36090  brcart  36128  dfrecs2  36148  dfrdg4  36149  cnambfre  38003  wepwsolem  43488  dnwech  43494  rankrelp  45405