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Theorem cossssid3 37981
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 37980 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
2 19.23v 1937 . . . . 5 (∀𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
32albii 1813 . . . 4 (∀𝑦𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
4 alcom 2148 . . . 4 (∀𝑦𝑢((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
53, 4bitr3i 276 . . 3 (∀𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
65albii 1813 . 2 (∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
7 alcom 2148 . 2 (∀𝑥𝑢𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
81, 6, 73bitri 296 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wex 1773  wss 3949   class class class wbr 5152   I cid 5579  ccoss 37689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-id 5580  df-coss 37923
This theorem is referenced by:  cossssid4  37982  cosscnvssid3  37988  cosselcnvrefrels3  38051  dffunALTV3  38201
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