Theorem epeli 5554

Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5553. (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 5553	. 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2145  Vcvv 3457   class class class wbr 5105   E cep 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-eprel 5552

This theorem is referenced by:  epel  5555  0sn0ep  5556  smoiso  8337  smoiso2  8344  ecid  8766  ordiso2  9465  cantnflt  9629  cantnfp1lem3  9637  oemapso  9639  cantnflem1b  9643  cantnflem1  9646  cantnf  9650  wemapwe  9654  cnfcomlem  9656  cnfcom  9657  cnfcom3lem  9660  leweon  9983  r0weon  9984  alephiso  10070  fin23lem27  10300  fpwwe2lem8  10611  oniso  28422  ex-eprel  30693  cardpred  35398  vonf1osev  35467  satefvfmla0  35781  satefvfmla1  35788  dftr6  36114  coep  36115  coepr  36116  brsset  36250  brtxpsd  36255  brcart  36293  dfrecs2  36313  dfrdg4  36314  cnambfre  38179  wepwsolem  43631  dnwech  43637  rankrelp  45534