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Theorem dfdisj2 4555
 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 4554 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 df-rmo 2904 . . 3 (∃*𝑥𝐴 𝑦𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦𝐵))
32albii 1737 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
41, 3bitri 263 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  ∃*wmo 2459  ∃*wrmo 2899  Disj wdisj 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-rmo 2904  df-disj 4554 This theorem is referenced by:  disjss1  4559  nfdisj  4565  invdisj  4571  sndisj  4574  disjxsn  4576  disjss3  4582  vitalilem3  23185
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