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Theorem dfdisj2 5024
 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 5023 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 df-rmo 3144 . . 3 (∃*𝑥𝐴 𝑦𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦𝐵))
32albii 1814 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
41, 3bitri 277 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1529   ∈ wcel 2108  ∃*wmo 2614  ∃*wrmo 3139  Disj wdisj 5022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804 This theorem depends on definitions:  df-bi 209  df-rmo 3144  df-disj 5023 This theorem is referenced by:  disjss1  5028  nfdisjw  5034  nfdisj  5035  invdisj  5041  sndisj  5048  disjxsn  5050  disjss3  5056  vitalilem3  24203
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