Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfdisjs2 Structured version   Visualization version   GIF version

Theorem dfdisjs2 35828
Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
dfdisjs2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ⊆ I }

Proof of Theorem dfdisjs2
StepHypRef Expression
1 dfdisjs 35827 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }
2 cosscnvelrels 35623 . . . 4 (𝑟 ∈ Rels → ≀ 𝑟 ∈ Rels )
32biantrud 532 . . 3 (𝑟 ∈ Rels → ( ≀ 𝑟 ⊆ I ↔ ( ≀ 𝑟 ⊆ I ∧ ≀ 𝑟 ∈ Rels )))
4 cosselcnvrefrels2 35660 . . 3 ( ≀ 𝑟 ∈ CnvRefRels ↔ ( ≀ 𝑟 ⊆ I ∧ ≀ 𝑟 ∈ Rels ))
53, 4syl6rbbr 291 . 2 (𝑟 ∈ Rels → ( ≀ 𝑟 ∈ CnvRefRels ↔ ≀ 𝑟 ⊆ I ))
61, 5rabimbieq 35400 1 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ⊆ I }
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1530  wcel 2107  {crab 3147  wss 3940   I cid 5458  ccnv 5553  ccoss 35340   Rels crels 35342   CnvRefRels ccnvrefrels 35348   Disjs cdisjs 35373
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-coss 35545  df-rels 35611  df-ssr 35624  df-cnvrefs 35649  df-cnvrefrels 35650  df-disjss 35822  df-disjs 35823
This theorem is referenced by:  dfdisjs3  35829  dfdisjs4  35830  dfdisjs5  35831
  Copyright terms: Public domain W3C validator