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Theorem nelb 3193
 Description: A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.)
Assertion
Ref Expression
nelb 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nelb
StepHypRef Expression
1 df-ne 2953 . . . . 5 (𝑥𝐴 ↔ ¬ 𝑥 = 𝐴)
21ralbii 3098 . . . 4 (∀𝑥𝐵 𝑥𝐴 ↔ ∀𝑥𝐵 ¬ 𝑥 = 𝐴)
3 ralnex 3164 . . . 4 (∀𝑥𝐵 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥𝐵 𝑥 = 𝐴)
42, 3bitri 278 . . 3 (∀𝑥𝐵 𝑥𝐴 ↔ ¬ ∃𝑥𝐵 𝑥 = 𝐴)
5 risset 3192 . . 3 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
64, 5xchbinxr 339 . 2 (∀𝑥𝐵 𝑥𝐴 ↔ ¬ 𝐴𝐵)
76bicomi 227 1 𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-clel 2831  df-ne 2953  df-ral 3076  df-rex 3077 This theorem is referenced by:  inpr0  30395
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