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Mirrors > Home > MPE Home > Th. List > elnelall | Structured version Visualization version GIF version |
Description: A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
Ref | Expression |
---|---|
elnelall | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3047 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | pm2.24 124 | . 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝜑)) | |
3 | 1, 2 | syl5bi 245 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2110 ∉ wnel 3046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-nel 3047 |
This theorem is referenced by: xnn0lenn0nn0 12835 ge2nprmge4 16258 afv2orxorb 44392 |
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