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Theorem cosscnvssid3 38457
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 38450 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
2 alrot3 2157 . 2 (∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
3 brcnvg 5892 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
43el2v 3484 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
5 brcnvg 5892 . . . . . 6 ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥𝑅𝑣𝑣𝑅𝑥))
65el2v 3484 . . . . 5 (𝑥𝑅𝑣𝑣𝑅𝑥)
74, 6anbi12i 628 . . . 4 ((𝑥𝑅𝑢𝑥𝑅𝑣) ↔ (𝑢𝑅𝑥𝑣𝑅𝑥))
87imbi1i 349 . . 3 (((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
983albii 1817 . 2 (∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
101, 2, 93bitri 297 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534  Vcvv 3477  wss 3962   class class class wbr 5147   I cid 5581  ccnv 5687  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-cnv 5696  df-coss 38392
This theorem is referenced by:  dfdisjs3  38691  dfdisjALTV3  38696  eldisjs3  38705
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