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Theorem disjxsn 5023
Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjxsn Disj 𝑥 ∈ {𝐴}𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjxsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 4997 . 2 (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
2 moeq 3606 . . 3 ∃*𝑥 𝑥 = 𝐴
3 elsni 4533 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 484 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) → 𝑥 = 𝐴)
54moimi 2545 . . 3 (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
62, 5ax-mp 5 . 2 ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵)
71, 6mpgbir 1806 1 Disj 𝑥 ∈ {𝐴}𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2114  ∃*wmo 2538  {csn 4516  Disj wdisj 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-mo 2540  df-clab 2717  df-cleq 2730  df-clel 2811  df-rmo 3061  df-sn 4517  df-disj 4996
This theorem is referenced by:  disjx0  5024  disjdifprg  30488  rossros  31718  meadjun  43542
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