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Theorem disjxsn 4838
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 4814 . 2 (Disj 𝑥 ∈ {𝐴}𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
2 moeq 3574 . . 3 ∃*𝑥 𝑥 = 𝐴
3 elsni 4387 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
43adantr 468 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦𝐵) → 𝑥 = 𝐴)
54moimi 2683 . . 3 (∃*𝑥 𝑥 = 𝐴 → ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵))
62, 5ax-mp 5 . 2 ∃*𝑥(𝑥 ∈ {𝐴} ∧ 𝑦𝐵)
71, 6mpgbir 1881 1 Disj 𝑥 ∈ {𝐴}𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wcel 2156  ∃*wmo 2631  {csn 4370  Disj wdisj 4812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rmo 3104  df-v 3393  df-sn 4371  df-disj 4813
This theorem is referenced by:  disjx0  4839  disjdifprg  29712  rossros  30567  meadjun  41155
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