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Mirrors > Home > MPE Home > Th. List > Mathboxes > trcoss | Structured version Visualization version GIF version |
Description: Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.) |
Ref | Expression |
---|---|
trcoss | ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moantr 38320 | . . . . 5 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ((∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧)) → ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧))) | |
2 | brcoss 38387 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))) | |
3 | 2 | el2v 3495 | . . . . . 6 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
4 | brcoss 38387 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧))) | |
5 | 4 | el2v 3495 | . . . . . 6 ⊢ (𝑦 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧)) |
6 | 3, 5 | anbi12i 627 | . . . . 5 ⊢ ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) ↔ (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧))) |
7 | brcoss 38387 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧))) | |
8 | 7 | el2v 3495 | . . . . 5 ⊢ (𝑥 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧)) |
9 | 1, 6, 8 | 3imtr4g 296 | . . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
10 | 9 | alrimiv 1926 | . . 3 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
11 | 10 | alimi 1809 | . 2 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
12 | 11 | alrimiv 1926 | 1 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 ∃wex 1777 ∃*wmo 2541 Vcvv 3488 class class class wbr 5166 ≀ ccoss 38135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-coss 38367 |
This theorem is referenced by: disjim 38737 |
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