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Theorem sndisj 5138
Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
sndisj Disj 𝑥𝐴 {𝑥}

Proof of Theorem sndisj
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 5114 . 2 (Disj 𝑥𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
2 moeq 3702 . . 3 ∃*𝑥 𝑥 = 𝑦
3 simpr 483 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4 velsn 4643 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
53, 4sylib 217 . . . . 5 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
65equcomd 2020 . . . 4 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
76moimi 2537 . . 3 (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
82, 7ax-mp 5 . 2 ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥})
91, 8mpgbir 1799 1 Disj 𝑥𝐴 {𝑥}
Colors of variables: wff setvar class
Syntax hints:  wa 394  wcel 2104  ∃*wmo 2530  {csn 4627  Disj wdisj 5112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-rmo 3374  df-v 3474  df-sn 4628  df-disj 5113
This theorem is referenced by:  0disj  5139  fnpreimac  32163  elrspunidl  32820  sibfof  33637  disjsnxp  44058  vonct  45707
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