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Theorem cosscnvssid3 35892
 Description: Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.)
Assertion
Ref Expression
cosscnvssid3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Distinct variable group:   𝑢,𝑅,𝑣,𝑥

Proof of Theorem cosscnvssid3
StepHypRef Expression
1 cossssid3 35885 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
2 alrot3 2161 . 2 (∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
3 brcnvg 5714 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
43el2v 3448 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
5 brcnvg 5714 . . . . . 6 ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥𝑅𝑣𝑣𝑅𝑥))
65el2v 3448 . . . . 5 (𝑥𝑅𝑣𝑣𝑅𝑥)
74, 6anbi12i 629 . . . 4 ((𝑥𝑅𝑢𝑥𝑅𝑣) ↔ (𝑢𝑅𝑥𝑣𝑅𝑥))
87imbi1i 353 . . 3 (((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
983albii 35685 . 2 (∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
101, 2, 93bitri 300 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   I cid 5424  ◡ccnv 5518   ≀ ccoss 35629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-cnv 5527  df-coss 35835 This theorem is referenced by:  dfdisjs3  36119  dfdisjALTV3  36124  eldisjs3  36133
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