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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 2804 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqneltrrd.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | |
4 | 2, 3 | eqneltrd 2909 | 1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: bitsf1 15785 lssvancl2 19710 lbsind2 19846 lindfind2 20507 2atjlej 36775 2atnelvolN 36883 lmod1zrnlvec 44903 |
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