Theorem resnonrel 43833

Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
resnonrel	⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Proof of Theorem resnonrel

Step	Hyp	Ref	Expression
1		ssv 3958	. . . 4 ⊢ 𝐵 ⊆ V
2		ssres2 5963	. . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)
4		cnvnonrel 43829	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	cnveqi 5823	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅
6		cnvcnv2 6151	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V)
7		cnv0 6097	. . . 4 ⊢ ◡∅ = ∅
8	5, 6, 7	3eqtr3i 2767	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅
9	3, 8	sseqtri 3982	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅
10		ss0b 4353	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅)
11	9, 10	mpbi 230	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Syntax hints:   = wceq 1541  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  ^◡ccnv 5623   ↾ cres 5626

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-res 5636

This theorem is referenced by:  imanonrel  43834