Theorem elnonrel 43581

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 43580	. . 3 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))
2	1	eleq2i 2821	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)))
3		eldif 3927	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ (V × V)) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴 ∧ ¬ ⟨𝑋, 𝑌⟩ ∈ (V × V)))
4		opelxp 5677	. . . . . 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 4863	. . . . . 6 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨𝑋, 𝑌⟩ = ∅)
8	7	eleq1d 2814	. . . . 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 2109  Vcvv 3450   ∖ cdif 3914  ∅c0 4299  ⟨cop 4598   × cxp 5639  ^◡ccnv 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649

This theorem is referenced by: (None)