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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 2827 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
| 4 | 1, 3 | mtbid 324 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 |
| This theorem is referenced by: neleqtrrd 2864 smoord 8405 r1tskina 10822 ofccat 15008 mreexexlem2d 17688 opptgdim2 28753 acopyeu 28842 dimlssid 33683 dochnel 41395 stoweidlem26 46041 fourierdlem60 46181 fourierdlem61 46182 sge00 46391 sge0sn 46394 sge0split 46424 |
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