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Theorem cosscnvssid3 38477
Description: Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.)
Assertion
Ref Expression
cosscnvssid3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Distinct variable group:   𝑢,𝑅,𝑣,𝑥

Proof of Theorem cosscnvssid3
StepHypRef Expression
1 cossssid3 38470 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
2 alrot3 2160 . 2 (∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
3 brcnvg 5890 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
43el2v 3487 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
5 brcnvg 5890 . . . . . 6 ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥𝑅𝑣𝑣𝑅𝑥))
65el2v 3487 . . . . 5 (𝑥𝑅𝑣𝑣𝑅𝑥)
74, 6anbi12i 628 . . . 4 ((𝑥𝑅𝑢𝑥𝑅𝑣) ↔ (𝑢𝑅𝑥𝑣𝑅𝑥))
87imbi1i 349 . . 3 (((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
983albii 1821 . 2 (∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
101, 2, 93bitri 297 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  Vcvv 3480  wss 3951   class class class wbr 5143   I cid 5577  ccnv 5684  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-cnv 5693  df-coss 38412
This theorem is referenced by:  dfdisjs3  38711  dfdisjALTV3  38716  eldisjs3  38725
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