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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelun | Structured version Visualization version GIF version |
Description: Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
Ref | Expression |
---|---|
nelun | ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2900 | . . . 4 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶))) | |
2 | elun 4118 | . . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | |
3 | 1, 2 | syl6bb 289 | . . 3 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))) |
4 | 3 | notbid 320 | . 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ ¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))) |
5 | ioran 980 | . 2 ⊢ (¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)) | |
6 | 4, 5 | syl6bb 289 | 1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∪ cun 3927 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-un 3934 |
This theorem is referenced by: cycpmco2 30794 |
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