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Theorem disjx0 4571
Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjx0 Disj 𝑥 ∈ ∅ 𝐵

Proof of Theorem disjx0
StepHypRef Expression
1 0ss 3923 . 2 ∅ ⊆ {∅}
2 disjxsn 4570 . 2 Disj 𝑥 ∈ {∅}𝐵
3 disjss1 4553 . 2 (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵Disj 𝑥 ∈ ∅ 𝐵))
41, 2, 3mp2 9 1 Disj 𝑥 ∈ ∅ 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3539  c0 3873  {csn 4124  Disj wdisj 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rmo 2903  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-sn 4125  df-disj 4548
This theorem is referenced by: (None)
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