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Theorem dfss6 3889
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 3886 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
2 notnotb 318 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2bitri 278 . 2 (𝐴𝐵 ↔ ¬ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
4 exanali 1867 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ¬ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4xchbinxr 338 1 (𝐴𝐵 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1541  wex 1787  wcel 2110  wss 3866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-ss 3883
This theorem is referenced by:  dfssr2  36354
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