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Theorem nssrex 39676
 Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Assertion
Ref Expression
nssrex 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nssrex
StepHypRef Expression
1 nss 3769 . 2 𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 df-rex 3020 . 2 (∃𝑥𝐴 ¬ 𝑥𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))
31, 2bitr4i 267 1 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383  ∃wex 1817   ∈ wcel 2103  ∃wrex 3015   ⊆ wss 3680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-rex 3020  df-in 3687  df-ss 3694 This theorem is referenced by:  mapssbi  39821
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