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Mirrors > Home > MPE Home > Th. List > nelaneq | Structured version Visualization version GIF version |
Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9062 analogously to elnotel 9073 and en2lp 9069. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) |
Ref | Expression |
---|---|
nelaneq | ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elneq 9062 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | |
2 | orc 863 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
3 | neneq 3022 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | |
4 | 3 | olcd 870 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
5 | 2, 4 | ja 188 | . . 3 ⊢ ((𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵) |
7 | ianor 978 | . 2 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | 6, 7 | mpbir 233 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-reg 9056 |
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-ne 3017 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4568 df-pr 4570 |
This theorem is referenced by: epinid0 9064 |
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