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Mirrors > Home > MPE Home > Th. List > Mathboxes > trcoss2 | Structured version Visualization version GIF version |
Description: Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.) |
Ref | Expression |
---|---|
trcoss2 | ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcom 2148 | . . 3 ⊢ (∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | |
2 | 1 | albii 1813 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
3 | 19.23v 1937 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) | |
4 | eleccossin 37865 | . . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))) | |
5 | 4 | el2v 3476 | . . . . . . 7 ⊢ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧)) |
6 | 5 | bicomi 223 | . . . . . 6 ⊢ ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) ↔ 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅)) |
7 | brcoss3 37815 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ 𝑅𝑧 ↔ ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) | |
8 | 7 | el2v 3476 | . . . . . 6 ⊢ (𝑥 ≀ 𝑅𝑧 ↔ ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅) |
9 | 6, 8 | imbi12i 350 | . . . . 5 ⊢ (((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
10 | 9 | albii 1813 | . . . 4 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
11 | n0 4341 | . . . . 5 ⊢ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅)) | |
12 | 11 | imbi1i 349 | . . . 4 ⊢ ((([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
13 | 3, 10, 12 | 3bitr4i 303 | . . 3 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
14 | 13 | 2albii 1814 | . 2 ⊢ (∀𝑥∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
15 | 2, 14 | bitri 275 | 1 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∩ cin 3942 ∅c0 4317 class class class wbr 5141 ◡ccnv 5668 [cec 8700 ≀ ccoss 37555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8704 df-coss 37793 |
This theorem is referenced by: eqvrelcoss4 38002 |
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