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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ecinn0 | Structured version Visualization version GIF version |
Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.) |
Ref | Expression |
---|---|
ecinn0 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecin0 37024 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥))) | |
2 | 1 | necon3abid 2976 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥))) |
3 | notnotb 314 | . . . . 5 ⊢ (𝐵𝑅𝑥 ↔ ¬ ¬ 𝐵𝑅𝑥) | |
4 | 3 | anbi2i 623 | . . . 4 ⊢ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥)) |
5 | 4 | exbii 1850 | . . 3 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥)) |
6 | exanali 1862 | . . 3 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ ¬ ¬ 𝐵𝑅𝑥) ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)) | |
7 | 5, 6 | bitri 274 | . 2 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ¬ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)) |
8 | 2, 7 | bitr4di 288 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 ∩ cin 3943 ∅c0 4318 class class class wbr 5141 [cec 8684 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8688 |
This theorem is referenced by: disjecxrn 37062 brcoss3 37106 brcosscnv2 37146 |
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