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Mirrors > Home > MPE Home > Th. List > neleqtrd | Structured version Visualization version GIF version |
Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
neleqtrd.1 | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
neleqtrd.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
neleqtrd | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neleqtrd.1 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | |
2 | neleqtrd.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2811 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
4 | 1, 3 | mtbid 323 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-clel 2802 |
This theorem is referenced by: neleqtrrd 2848 smoord 8386 r1tskina 10812 ofccat 14960 mreexexlem2d 17644 opptgdim2 28641 acopyeu 28730 dochnel 41016 stoweidlem26 45557 fourierdlem60 45697 fourierdlem61 45698 sge00 45907 sge0sn 45910 sge0split 45940 |
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