Theorem epeli 5468

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5466. (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 5466	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2114  Vcvv 3494   class class class wbr 5066   E cep 5464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-eprel 5465

This theorem is referenced by:  epel  5469  0sn0ep  5470  epini  5959  smoiso  7999  smoiso2  8006  ecid  8362  ordiso2  8979  cantnflt  9135  cantnfp1lem3  9143  oemapso  9145  cantnflem1b  9149  cantnflem1  9152  cantnf  9156  wemapwe  9160  cnfcomlem  9162  cnfcom  9163  cnfcom3lem  9166  leweon  9437  r0weon  9438  alephiso  9524  fin23lem27  9750  fpwwe2lem9  10060  ex-eprel  28212  satefvfmla0  32665  satefvfmla1  32672  dftr6  32986  coep  32987  coepr  32988  brsset  33350  brtxpsd  33355  brcart  33393  dfrecs2  33411  dfrdg4  33412  cnambfre  34955  wepwsolem  39691  dnwech  39697