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Theorem eleccossin 38388
Description: Two ways of saying that the coset of 𝐴 and the coset of 𝐶 have the common element 𝐵. (Contributed by Peter Mazsa, 15-Oct-2021.)
Assertion
Ref Expression
eleccossin ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))

Proof of Theorem eleccossin
StepHypRef Expression
1 elin 3986 . . 3 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅))
2 relcoss 38328 . . . . 5 Rel ≀ 𝑅
3 relelec 8806 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵))
42, 3ax-mp 5 . . . 4 (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵)
5 relelec 8806 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵))
62, 5ax-mp 5 . . . 4 (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵)
74, 6anbi12i 627 . . 3 ((𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
81, 7bitri 275 . 2 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
9 brcosscnvcoss 38339 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵𝑅𝐶𝐶𝑅𝐵))
109anbi2d 629 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐴𝑅𝐵𝐵𝑅𝐶) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵)))
118, 10bitr4id 290 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2103  cin 3969   class class class wbr 5169  Rel wrel 5704  [cec 8757  ccoss 38084
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ec 8761  df-coss 38316
This theorem is referenced by:  trcoss2  38389
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