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Theorem cossssid3 36977
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.)
Assertion
Ref Expression
cossssid3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cossssid3
StepHypRef Expression
1 cossssid2 36976 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
2 19.23v 1946 . . . . 5 (∀𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
32albii 1822 . . . 4 (∀𝑦𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
4 alcom 2157 . . . 4 (∀𝑦𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
53, 4bitr3i 277 . . 3 (∀𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
65albii 1822 . 2 (∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
7 alcom 2157 . 2 (∀𝑥𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
81, 6, 73bitri 297 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wss 3911   class class class wbr 5106   I cid 5531  ccoss 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5169  df-id 5532  df-coss 36919
This theorem is referenced by:  cossssid4  36978  cosscnvssid3  36984  cosselcnvrefrels3  37047  dffunALTV3  37197
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