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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 9046 analogously to elnotel 9057 and en2lp 9053. (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 9046 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | |
2 | orc 864 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
3 | neneq 2993 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | |
4 | 3 | olcd 871 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
5 | 2, 4 | ja 189 | . . 3 ⊢ ((𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵) |
7 | ianor 979 | . 2 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | 6, 7 | mpbir 234 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-nul 4244 df-sn 4526 df-pr 4528 |
This theorem is referenced by: epinid0 9048 |
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