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

Theorem dfdisj2 4546
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 4545 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 df-rmo 2900 . . 3 (∃*𝑥𝐴 𝑦𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦𝐵))
32albii 1736 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
41, 3bitri 262 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wal 1472  wcel 1976  ∃*wmo 2455  ∃*wrmo 2895  Disj wdisj 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-rmo 2900  df-disj 4545
This theorem is referenced by:  disjss1  4550  nfdisj  4556  invdisj  4562  sndisj  4565  disjxsn  4567  disjss3  4573  vitalilem3  23099
  Copyright terms: Public domain W3C validator