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Theorem eleccossin 38439
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 3992 . . 3 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅))
2 relcoss 38379 . . . . 5 Rel ≀ 𝑅
3 relelec 8810 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵))
42, 3ax-mp 5 . . . 4 (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵)
5 relelec 8810 . . . . 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 38390 . . 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 2108  cin 3975   class class class wbr 5166  Rel wrel 5705  [cec 8761  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-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  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-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-coss 38367
This theorem is referenced by:  trcoss2  38440
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