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Theorem elnonrel 38208
Description: Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.)
Assertion
Ref Expression
elnonrel (⟨𝑋, 𝑌⟩ ∈ (𝐴𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

Proof of Theorem elnonrel
StepHypRef Expression
1 nonrel 38207 . . 3 (𝐴𝐴) = (𝐴 ∖ (V × V))
21eleq2i 2722 . 2 (⟨𝑋, 𝑌⟩ ∈ (𝐴𝐴) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)))
3 eldif 3617 . . 3 (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
4 opelxp 5180 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V))
54notbii 309 . . . . 5 (¬ ⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))
65anbi2i 730 . . . 4 ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
7 opprc 4456 . . . . . 6 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
87eleq1d 2715 . . . . 5 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (⟨𝑋, 𝑌⟩ ∈ 𝐴 ↔ ∅ ∈ 𝐴))
98pm5.32ri 671 . . . 4 ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
106, 9bitri 264 . . 3 ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
113, 10bitri 264 . 2 (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
122, 11bitri 264 1 (⟨𝑋, 𝑌⟩ ∈ (𝐴𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wcel 2030  Vcvv 3231  cdif 3604  c0 3948  cop 4216   × cxp 5141  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151
This theorem is referenced by: (None)
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