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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 9542 analogously to elnotel 9554 and en2lp 9550. (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 9542 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) | |
2 | orc 866 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
3 | neneq 2946 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵) | |
4 | 3 | olcd 873 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
5 | 2, 4 | ja 186 | . . 3 ⊢ ((𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵) |
7 | ianor 981 | . 2 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
8 | 6, 7 | mpbir 230 | 1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3449 df-un 3919 df-sn 4591 df-pr 4593 |
This theorem is referenced by: epinid0 9544 |
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