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Theorem eleccossin 34475
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 34431 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵𝑅𝐶𝐶𝑅𝐵))
21anbi2d 742 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐴𝑅𝐵𝐵𝑅𝐶) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵)))
3 elin 3904 . . 3 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅))
4 relcoss 34420 . . . . 5 Rel ≀ 𝑅
5 relelec 7905 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵))
64, 5ax-mp 5 . . . 4 (𝐵 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝐵)
7 relelec 7905 . . . . 5 (Rel ≀ 𝑅 → (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵))
84, 7ax-mp 5 . . . 4 (𝐵 ∈ [𝐶] ≀ 𝑅𝐶𝑅𝐵)
96, 8anbi12i 735 . . 3 ((𝐵 ∈ [𝐴] ≀ 𝑅𝐵 ∈ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
103, 9bitri 264 . 2 (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐶𝑅𝐵))
112, 10syl6rbbr 279 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2103  cin 3679   class class class wbr 4760  Rel wrel 5223  [cec 7860  ccoss 34215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ec 7864  df-coss 34411
This theorem is referenced by:  trcoss2  34476
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