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Theorem eleccossin 35842
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 35798 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵𝑅𝐶𝐶𝑅𝐵))
21anbi2d 631 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐴𝑅𝐵𝐵𝑅𝐶) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵)))
3 elin 3924 . . 3 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅))
4 relcoss 35787 . . . . 5 Rel ≀ 𝑅
5 relelec 8321 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵))
64, 5ax-mp 5 . . . 4 (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵)
7 relelec 8321 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵))
84, 7ax-mp 5 . . . 4 (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵)
96, 8anbi12i 629 . . 3 ((𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
103, 9bitri 278 . 2 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
112, 10syl6rbbr 293 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2114  cin 3907   class class class wbr 5042  Rel wrel 5537  [cec 8274  ccoss 35572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ec 8278  df-coss 35778
This theorem is referenced by:  trcoss2  35843
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