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Theorem elnelneqd 40827
Description: Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
elnelneqd.1 (𝜑𝐶𝐴)
elnelneqd.2 (𝜑 → ¬ 𝐶𝐵)
Assertion
Ref Expression
elnelneqd (𝜑 → ¬ 𝐴 = 𝐵)

Proof of Theorem elnelneqd
StepHypRef Expression
1 elnelneqd.2 . 2 (𝜑 → ¬ 𝐶𝐵)
2 elnelneqd.1 . . . 4 (𝜑𝐶𝐴)
32adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶𝐴)
4 simpr 488 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
53, 4eleqtrd 2918 . 2 ((𝜑𝐴 = 𝐵) → 𝐶𝐵)
61, 5mtand 815 1 (𝜑 → ¬ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896
This theorem is referenced by:  mnuprdlem1  40900  mnuprdlem2  40901
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