Theorem brnonrel 42803

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 5197	. 2 ⊢ ¬ 𝑌∅𝑋
2		brcnvg 5879	. . . 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 42802	. . . 4 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	breqi 5154	. . 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 205   ∧ wa 395   ∈ wcel 2105   ∖ cdif 3945  ∅c0 4322   class class class wbr 5148  ^◡ccnv 5675

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 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684

This theorem is referenced by: (None)