Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndisj2 | Structured version Visualization version GIF version |
Description: A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
ndisj2.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
ndisj2 | ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndisj2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
2 | 1 | disjor 5038 | . . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅)) |
3 | 2 | notbii 322 | . 2 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅)) |
4 | rexnal 3238 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅)) | |
5 | rexnal 3238 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅)) | |
6 | ioran 980 | . . . . . 6 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅)) | |
7 | df-ne 3017 | . . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
8 | df-ne 3017 | . . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐵 ∩ 𝐶) = ∅) | |
9 | 7, 8 | anbi12i 628 | . . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅)) |
10 | 6, 9 | bitr4i 280 | . . . . 5 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
11 | 10 | rexbii 3247 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
12 | 5, 11 | bitr3i 279 | . . 3 ⊢ (¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
13 | 12 | rexbii 3247 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
14 | 3, 4, 13 | 3bitr2i 301 | 1 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ∅c0 4290 Disj wdisj 5023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rmo 3146 df-v 3496 df-dif 3938 df-in 3942 df-nul 4291 df-disj 5024 |
This theorem is referenced by: disjrnmpt2 41442 |
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