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Theorem cossssid2 34533
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 5226 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
21sseq2i 3834 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
3 df-coss 34484 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
43sseq1i 3833 . 2 ( ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
5 ssopab2b 5204 . 2 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
62, 4, 53bitri 288 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wss 3776   class class class wbr 4851  {copab 4913   I cid 5225  ccoss 34295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-opab 4914  df-id 5226  df-coss 34484
This theorem is referenced by:  cossssid3  34534
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