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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 2820 | . 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 1542 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-clel 2811 |
This theorem is referenced by: neleqtrrd 2857 smoord 8312 r1tskina 10723 ofccat 14860 mreexexlem2d 17530 opptgdim2 27729 acopyeu 27818 dochnel 39902 stoweidlem26 44353 fourierdlem60 44493 fourierdlem61 44494 sge00 44703 sge0sn 44706 sge0split 44736 |
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