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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 2898 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) |
4 | 1, 3 | mtbid 326 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
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: neleqtrrd 2935 smoord 8002 r1tskina 10204 ofccat 14329 mreexexlem2d 16916 opptgdim2 26531 acopyeu 26620 dochnel 38544 stoweidlem26 42331 fourierdlem60 42471 fourierdlem61 42472 sge00 42678 sge0sn 42681 sge0split 42711 iundjiunlem 42761 |
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