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Mirrors > Home > MPE Home > Th. List > eldifsnneqOLD | Structured version Visualization version GIF version |
Description: Obsolete version of eldifsnneq 4723 as of 1-Jun-2023. 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.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
eldifsnneqOLD | ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3946 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶})) | |
2 | elsng 4581 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶)) | |
3 | 2 | biimprd 250 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 = 𝐶 → 𝐴 ∈ {𝐶})) |
4 | 3 | con3dimp 411 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶}) → ¬ 𝐴 = 𝐶) |
5 | 1, 4 | sylbi 219 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → ¬ 𝐴 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 {csn 4567 |
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-v 3496 df-dif 3939 df-sn 4568 |
This theorem is referenced by: (None) |
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