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Theorem disjxsn 5136
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 5111 . 2 (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
2 moeq 3712 . . 3 ∃*𝑥 𝑥 = 𝐴
3 elsni 4642 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 480 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) → 𝑥 = 𝐴)
54moimi 2544 . . 3 (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
62, 5ax-mp 5 . 2 ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵)
71, 6mpgbir 1798 1 Disj 𝑥 ∈ {𝐴}𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  ∃*wmo 2537  {csn 4625  Disj wdisj 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rmo 3379  df-sn 4626  df-disj 5110
This theorem is referenced by:  disjx0  5137  disjdifprg  32589  rossros  34182  meadjun  46482
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