Theorem brnonrel 39942

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 5107	. 2 ⊢ ¬ 𝑌∅𝑋
2		cnvnonrel 39941	. . . 4 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	breqi 5064	. . 3 ⊢ (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑌∅𝑋)
4		brcnvg 5744	. . . 4 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑈) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌))
5	4	ancoms 461	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑌◡(𝐴 ∖ ◡◡𝐴)𝑋 ↔ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌))
6	3, 5	syl5rbbr 288	. 2 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → (𝑋(𝐴 ∖ ◡◡𝐴)𝑌 ↔ 𝑌∅𝑋))
7	1, 6	mtbiri 329	1 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   ∖ cdif 3932  ∅c0 4290   class class class wbr 5058  ^◡ccnv 5548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557

This theorem is referenced by: (None)