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Theorem disjxsn 5145
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 5119 . 2 (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
2 moeq 3704 . . 3 ∃*𝑥 𝑥 = 𝐴
3 elsni 4649 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 479 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) → 𝑥 = 𝐴)
54moimi 2534 . . 3 (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
62, 5ax-mp 5 . 2 ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵)
71, 6mpgbir 1793 1 Disj 𝑥 ∈ {𝐴}𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  ∃*wmo 2527  {csn 4632  Disj wdisj 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-rmo 3374  df-sn 4633  df-disj 5118
This theorem is referenced by:  disjx0  5146  disjdifprg  32386  rossros  33832  meadjun  45879
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