Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eleccossin Structured version   Visualization version   GIF version

Theorem eleccossin 35603
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 brcosscnvcoss 35559 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵𝑅𝐶𝐶𝑅𝐵))
21anbi2d 628 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐴𝑅𝐵𝐵𝑅𝐶) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵)))
3 elin 4166 . . 3 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅))
4 relcoss 35548 . . . . 5 Rel ≀ 𝑅
5 relelec 8323 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵))
64, 5ax-mp 5 . . . 4 (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵)
7 relelec 8323 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵))
84, 7ax-mp 5 . . . 4 (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵)
96, 8anbi12i 626 . . 3 ((𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
103, 9bitri 276 . 2 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
112, 10syl6rbbr 291 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  cin 3932   class class class wbr 5057  Rel wrel 5553  [cec 8276  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-coss 35539
This theorem is referenced by:  trcoss2  35604
  Copyright terms: Public domain W3C validator