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Mirrors > Home > MPE Home > Th. List > eqneltrrd | Structured version Visualization version GIF version |
Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) |
Ref | Expression |
---|---|
eqneltrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqneltrrd.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Ref | Expression |
---|---|
eqneltrrd | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqneltrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eqcomd 2827 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqneltrrd.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | |
4 | 2, 3 | eqneltrd 2932 | 1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: bitsf1 15795 lssvancl2 19717 lbsind2 19853 lindfind2 20962 2atjlej 36630 2atnelvolN 36738 lmod1zrnlvec 44569 |
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