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Theorem dfss6 3751
Description: Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
dfss6 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss6
StepHypRef Expression
1 dfss2 3749 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 notnotb 306 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 266 . 2 (𝐴𝐵 ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
4 exanali 1955 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4xchbinxr 326 1 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  wss 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-in 3739  df-ss 3746
This theorem is referenced by:  dfssr2  34610
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