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Mirrors > Home > MPE Home > Th. List > elneeldif | Structured version Visualization version GIF version |
Description: The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.) |
Ref | Expression |
---|---|
elneeldif | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3946 | . . 3 ⊢ (𝑌 ∈ (𝐵 ∖ 𝐴) ↔ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴)) | |
2 | nelne2 3115 | . . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌) | |
3 | 2 | ex 415 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑌 ∈ 𝐴 → 𝑋 ≠ 𝑌)) |
4 | 3 | adantld 493 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌)) |
5 | 1, 4 | syl5bi 244 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ (𝐵 ∖ 𝐴) → 𝑋 ≠ 𝑌)) |
6 | 5 | imp 409 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 |
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-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3939 |
This theorem is referenced by: fmlasucdisj 32646 |
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