Theorem brnonrel 43571

Description: A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)

Assertion

Ref	Expression
brnonrel	⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)

Proof of Theorem brnonrel

Step	Hyp	Ref	Expression
1		br0 5158	. 2 ⊢ ¬ 𝑌∅𝑋
2		brcnvg 5845	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑈) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌))
3	2	ancoms 458	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌))
4		cnvnonrel 43570	. . . 4 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	breqi 5115	. . 3 ⊢ (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑌∅𝑋)
6	3, 5	bitr3di 286	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑋(𝐴 ∖ ◡◡𝐴)𝑌 ↔ 𝑌∅𝑋))
7	1, 6	mtbiri 327	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∖ cdif 3913  ∅c0 4298   class class class wbr 5109  ^◡ccnv 5639

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 5253  ax-nul 5263  ax-pr 5389

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-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648

This theorem is referenced by: (None)