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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 4793 | . 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 1541 ∈ wcel 2106 ∖ cdif 3945 {csn 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-v 3476 df-dif 3951 df-sn 4629 |
This theorem is referenced by: mpodifsnif 7522 symgextfv 19285 evlslem3 21642 evlslem1 21644 2sqreultblem 26948 qsdrngi 32604 elzdif0 32955 bj-fvsnun1 36131 evlsbagval 41140 selvvvval 41159 clsk3nimkb 42781 fdmdifeqresdif 47007 |
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