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Theorem cossssid4 34535
 Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.)
Assertion
Ref Expression
cossssid4 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
Distinct variable group:   𝑢,𝑅,𝑥

Proof of Theorem cossssid4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cossssid3 34534 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
2 breq2 4800 . . . 4 (𝑥 = 𝑦 → (𝑢𝑅𝑥𝑢𝑅𝑦))
32mo4 2647 . . 3 (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
43albii 1888 . 2 (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
51, 4bitr4i 267 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1622   = wceq 1624  ∃*wmo 2600   ⊆ wss 3707   class class class wbr 4796   I cid 5165   ≀ ccoss 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-coss 34484 This theorem is referenced by:  cossssid5  34536  cosscnvssid4  34542  cosselcnvrefrels4  34601
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