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Theorem dfdisj2 3776
 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 3775 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
2 df-rmo 2357 . . 3 (∃*𝑥𝐴 𝑦𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦𝐵))
32albii 1400 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
41, 3bitri 182 1 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   ↔ wb 103  ∀wal 1283   ∈ wcel 1434  ∃*wmo 1943  ∃*wrmo 2352  Disj wdisj 3774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379 This theorem depends on definitions:  df-bi 115  df-rmo 2357  df-disj 3775 This theorem is referenced by:  disjss1  3780  nfdisjv  3786  invdisj  3788  sndisj  3789  disjxsn  3791
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