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Theorem sndisj 5022
 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 4998 . 2 (Disj 𝑥𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
2 moeq 3646 . . 3 ∃*𝑥 𝑥 = 𝑦
3 simpr 488 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4 velsn 4541 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
53, 4sylib 221 . . . . 5 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
65equcomd 2026 . . . 4 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
76moimi 2603 . . 3 (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
82, 7ax-mp 5 . 2 ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥})
91, 8mpgbir 1801 1 Disj 𝑥𝐴 {𝑥}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∈ wcel 2111  ∃*wmo 2596  {csn 4525  Disj wdisj 4996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-rmo 3114  df-v 3443  df-sn 4526  df-disj 4997 This theorem is referenced by:  0disj  5023  fnpreimac  30448  elrspunidl  31045  sibfof  31744  disjsnxp  41768  vonct  43393
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