Theorem elnonrel 44544

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 44543	. . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))
2	1	eleq2i 2853	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)))
3		eldif 3909	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
4		opelxp 5687	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V))
5	4	notbii 323	. . . . 5 ⊢ (¬ ⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V))
6	5	anbi2i 635	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
7		opprc 4856	. . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
8	7	eleq1d 2846	. . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (⟨𝑋, 𝑌⟩ ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	pm5.32ri 586	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
10	6, 9	bitri 278	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
11	3, 10	bitri 278	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
12	2, 11	bitri 278	1 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896  ∅c0 4279  ⟨cop 4590   × cxp 5649  ^◡ccnv 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659

This theorem is used by: (None)