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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 2853 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
4 | 1, 3 | mtbid 316 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1508 ∈ wcel 2051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-ext 2752 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-cleq 2773 df-clel 2848 |
This theorem is referenced by: neleqtrrd 2890 smoord 7812 r1tskina 10008 ofccat 14196 mreexexlem2d 16786 opptgdim2 26248 acopyeu 26338 dochnel 38014 stoweidlem26 41777 fourierdlem60 41917 fourierdlem61 41918 sge00 42124 sge0sn 42127 sge0split 42157 iundjiunlem 42207 |
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