MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjxsn Structured version   Visualization version   GIF version

Theorem disjxsn 5050
Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjxsn Disj 𝑥 ∈ {𝐴}𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjxsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 5024 . 2 (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
2 moeq 3695 . . 3 ∃*𝑥 𝑥 = 𝐴
3 elsni 4574 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 481 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) → 𝑥 = 𝐴)
54moimi 2620 . . 3 (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
62, 5ax-mp 5 . 2 ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵)
71, 6mpgbir 1791 1 Disj 𝑥 ∈ {𝐴}𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  ∃*wmo 2613  {csn 4557  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rmo 3143  df-sn 4558  df-disj 5023
This theorem is referenced by:  disjx0  5051  disjdifprg  30253  rossros  31338  meadjun  42621
  Copyright terms: Public domain W3C validator