Theorem elnonrel 43617

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⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

Proof of Theorem elnonrel

Step	Hyp	Ref	Expression
1		nonrel 43616	. . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))
2	1	eleq2i 2823	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)))
3		eldif 3912	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
4		opelxp 5652	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V))
5	4	notbii 320	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))
6	5	anbi2i 623	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
7		opprc 4848	. . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
8	7	eleq1d 2816	. . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (⟨𝑋, 𝑌⟩ ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	pm5.32ri 575	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
10	6, 9	bitri 275	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
11	3, 10	bitri 275	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
12	2, 11	bitri 275	1 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899  ∅c0 4283  ⟨cop 4582   × cxp 5614  ^◡ccnv 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624

This theorem is referenced by: (None)