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Theorem cossssid2 35189
 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
cossssid2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cossssid2
StepHypRef Expression
1 df-id 5340 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
21sseq2i 3912 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
3 df-coss 35140 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
43sseq1i 3911 . 2 ( ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
5 ssopab2b 5316 . 2 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
62, 4, 53bitri 298 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1518  ∃wex 1759   ⊆ wss 3854   class class class wbr 4956  {copab 5018   I cid 5339   ≀ ccoss 34933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-opab 5019  df-id 5340  df-coss 35140 This theorem is referenced by:  cossssid3  35190
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