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Theorem sndisj 3751
Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
sndisj Disj x A {x}

Proof of Theorem sndisj
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3738 . 2 (Disj x A {x} ↔ y∃*x(x A y {x}))
2 moeq 2710 . . 3 ∃*x x = y
3 simpr 103 . . . . . 6 ((x A y {x}) → y {x})
4 elsn 3382 . . . . . 6 (y {x} ↔ y = x)
53, 4sylib 127 . . . . 5 ((x A y {x}) → y = x)
65eqcomd 2042 . . . 4 ((x A y {x}) → x = y)
76moimi 1962 . . 3 (∃*x x = y∃*x(x A y {x}))
82, 7ax-mp 7 . 2 ∃*x(x A y {x})
91, 8mpgbir 1339 1 Disj x A {x}
Colors of variables: wff set class
Syntax hints:   wa 97   wcel 1390  ∃*wmo 1898  {csn 3367  Disj wdisj 3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-rmo 2308  df-v 2553  df-sn 3373  df-disj 3737
This theorem is referenced by:  0disj  3752
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