MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epn0 Structured version   Visualization version   GIF version

Theorem epn0 5576
Description: The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
epn0 E ≠ ∅

Proof of Theorem epn0
StepHypRef Expression
1 0sn0ep 5575 . 2 ∅ E {∅}
2 brne0 5189 . 2 (∅ E {∅} → E ≠ ∅)
31, 2ax-mp 5 1 E ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2932  c0 4315  {csn 4621   class class class wbr 5139   E cep 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571
This theorem is referenced by:  epnsym  9601
  Copyright terms: Public domain W3C validator