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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 4737 | . 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 2105 ∖ cdif 3895 {csn 4573 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-v 3443 df-dif 3901 df-sn 4574 |
This theorem is referenced by: mpodifsnif 7451 symgextfv 19122 evlslem3 21396 evlslem1 21398 2sqreultblem 26702 elzdif0 32228 bj-fvsnun1 35539 evlsbagval 40543 mhphf 40553 clsk3nimkb 41980 fdmdifeqresdif 46037 |
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