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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 4792 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶) | |
2 | 1 | neneqd 2943 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2104 ∖ cdif 3944 {csn 4627 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-v 3474 df-dif 3950 df-sn 4628 |
This theorem is referenced by: mpodifsnif 7525 symgextfv 19327 evlslem3 21862 evlslem1 21864 2sqreultblem 27187 qsdrngi 32883 elzdif0 33258 bj-fvsnun1 36439 evlsbagval 41440 selvvvval 41459 clsk3nimkb 43093 fdmdifeqresdif 47105 |
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