MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noxpordse Structured version   Visualization version   GIF version

Theorem noxpordse 27897
Description: Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
Hypotheses
Ref Expression
noxpord.1 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
noxpord.2 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑅(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
Assertion
Ref Expression
noxpordse 𝑆 Se ( No × No )
Distinct variable groups:   𝑥,𝑅,𝑦   𝑎,𝑏
Allowed substitution hints:   𝑅(𝑎,𝑏)   𝑆(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem noxpordse
StepHypRef Expression
1 noxpord.2 . . 3 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑅(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
2 noxpord.1 . . . . 5 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
32lrrecse 27887 . . . 4 𝑅 Se No
43a1i 11 . . 3 (⊤ → 𝑅 Se No )
51, 4, 4sexp2 8159 . 2 (⊤ → 𝑆 Se ( No × No ))
65mptru 1540 1 𝑆 Se ( No × No )
Colors of variables: wff setvar class
Syntax hints:  wo 845  w3a 1084   = wceq 1533  wtru 1534  wcel 2098  wne 2937  cun 3947   class class class wbr 5152  {copab 5214   Se wse 5635   × cxp 5680  cfv 6553  1st c1st 7999  2nd c2nd 8000   No csur 27601   L cleft 27800   R cright 27801
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-se 5638  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 8001  df-2nd 8002
This theorem is referenced by:  norec2fn  27901  norec2ov  27902
  Copyright terms: Public domain W3C validator