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| Mirrors > Home > MPE Home > Th. List > eldifsnneq | Structured version Visualization version GIF version | ||
| Description: An element of a difference with a singleton is not equal to the element of that singleton. Note that (¬ 𝐴 ∈ {𝐶} → ¬ 𝐴 = 𝐶) need not hold if 𝐴 is a proper class. (Contributed by BJ, 18-Mar-2023.) (Proof shortened by Steven Nguyen, 1-Jun-2023.) |
| Ref | Expression |
|---|---|
| eldifsnneq | ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldifsni 4790 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶) | |
| 2 | 1 | neneqd 2945 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 {csn 4626 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 |
| This theorem is referenced by: mpodifsnif 7548 symgextfv 19436 evlslem3 22104 evlslem1 22106 2sqreultblem 27492 qsdrngi 33523 elzdif0 33981 bj-fvsnun1 37256 evlsbagval 42576 selvvvval 42595 clsk3nimkb 44053 fdmdifeqresdif 48258 |
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