Theorem elnonrel 41566

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 41565	. . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))
2	1	eleq2i 2828	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)))
3		eldif 3908	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
4		opelxp 5657	. . . . . 6 ⊢ (⟨𝑋, 𝑌⟩ ∈ (V × V) ↔ (𝑋 ∈ V ∧ 𝑌 ∈ V))
5	4	notbii 319	. . . . 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 4841	. . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
8	7	eleq1d 2821	. . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (⟨𝑋, 𝑌⟩ ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	pm5.32ri 576	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
10	6, 9	bitri 274	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
11	3, 10	bitri 274	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
12	2, 11	bitri 274	1 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∈ wcel 2105  Vcvv 3441   ∖ cdif 3895  ∅c0 4270  ⟨cop 4580   × cxp 5619  ^◡ccnv 5620

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-br 5094  df-opab 5156  df-xp 5627  df-rel 5628  df-cnv 5629

This theorem is referenced by: (None)