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Theorem dfss6 3966
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 df-ss 3961 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 notnotb 314 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 274 . 2 (𝐴𝐵 ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
4 exanali 1854 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4xchbinxr 334 1 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1531  wex 1773  wcel 2098  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-ss 3961
This theorem is referenced by:  dfssr2  38121
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