MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdisj2 Structured version   Visualization version   GIF version

Theorem dfdisj2 5117
Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
dfdisj2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem dfdisj2
StepHypRef Expression
1 df-disj 5116 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 df-rmo 3378 . . 3 (∃*𝑥𝐴 𝑦𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦𝐵))
32albii 1816 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
41, 3bitri 275 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1535  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-rmo 3378  df-disj 5116
This theorem is referenced by:  disjss1  5121  nfdisjw  5127  nfdisj  5128  invdisj  5134  sndisj  5140  disjxsn  5142  disjss3  5147  vitalilem3  25659
  Copyright terms: Public domain W3C validator