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Theorem sndisj 5140
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 5116 . 2 (Disj 𝑥𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
2 moeq 3704 . . 3 ∃*𝑥 𝑥 = 𝑦
3 simpr 486 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4 velsn 4645 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
53, 4sylib 217 . . . . 5 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
65equcomd 2023 . . . 4 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
76moimi 2540 . . 3 (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
82, 7ax-mp 5 . 2 ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥})
91, 8mpgbir 1802 1 Disj 𝑥𝐴 {𝑥}
Colors of variables: wff setvar class
Syntax hints:  wa 397  wcel 2107  ∃*wmo 2533  {csn 4629  Disj wdisj 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-rmo 3377  df-v 3477  df-sn 4630  df-disj 5115
This theorem is referenced by:  0disj  5141  fnpreimac  31896  elrspunidl  32546  sibfof  33339  disjsnxp  43757  vonct  45409
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