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Theorem sndisj 3788
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 3775 . 2 (Disj 𝑥𝐴 {𝑥} ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
2 moeq 2739 . . 3 ∃*𝑥 𝑥 = 𝑦
3 simpr 107 . . . . . 6 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 ∈ {𝑥})
4 velsn 3420 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
53, 4sylib 131 . . . . 5 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑦 = 𝑥)
65eqcomd 2061 . . . 4 ((𝑥𝐴𝑦 ∈ {𝑥}) → 𝑥 = 𝑦)
76moimi 1981 . . 3 (∃*𝑥 𝑥 = 𝑦 → ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥}))
82, 7ax-mp 7 . 2 ∃*𝑥(𝑥𝐴𝑦 ∈ {𝑥})
91, 8mpgbir 1358 1 Disj 𝑥𝐴 {𝑥}
Colors of variables: wff set class
Syntax hints:  wa 101  wcel 1409  ∃*wmo 1917  {csn 3403  Disj wdisj 3773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-v 2576  df-sn 3409  df-disj 3774
This theorem is referenced by:  0disj  3789
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