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Theorem 0disj 4569
Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
0disj Disj 𝑥𝐴

Proof of Theorem 0disj
StepHypRef Expression
1 0ss 3923 . . 3 ∅ ⊆ {𝑥}
21rgenw 2907 . 2 𝑥𝐴 ∅ ⊆ {𝑥}
3 sndisj 4568 . 2 Disj 𝑥𝐴 {𝑥}
4 disjss2 4550 . 2 (∀𝑥𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥𝐴 {𝑥} → Disj 𝑥𝐴 ∅))
52, 3, 4mp2 9 1 Disj 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wral 2895  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 2032  ax-13 2232  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-ral 2900  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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