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Theorem sndisj 5044
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 5020 . 2 (Disj 𝑥𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
2 moeq 3620 . . 3 ∃*𝑥 𝑥 = 𝑦
3 simpr 488 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4 velsn 4557 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
53, 4sylib 221 . . . . 5 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
65equcomd 2027 . . . 4 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
76moimi 2544 . . 3 (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
82, 7ax-mp 5 . 2 ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥})
91, 8mpgbir 1807 1 Disj 𝑥𝐴 {𝑥}
Colors of variables: wff setvar class
Syntax hints:  wa 399  wcel 2110  ∃*wmo 2537  {csn 4541  Disj wdisj 5018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2816  df-rmo 3069  df-v 3410  df-sn 4542  df-disj 5019
This theorem is referenced by:  0disj  5045  fnpreimac  30728  elrspunidl  31320  sibfof  32019  disjsnxp  42291  vonct  43906
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