Theorem resnonrel 43712

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 3955	. . . 4 ⊢ 𝐵 ⊆ V
2		ssres2 5959	. . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)
4		cnvnonrel 43708	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	cnveqi 5820	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅
6		cnvcnv2 6147	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V)
7		cnv0 6093	. . . 4 ⊢ ◡∅ = ∅
8	5, 6, 7	3eqtr3i 2764	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅
9	3, 8	sseqtri 3979	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅
10		ss0b 4350	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅)
11	9, 10	mpbi 230	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Syntax hints:   = wceq 1541  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898  ∅c0 4282  ^◡ccnv 5620   ↾ cres 5623

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 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-res 5633

This theorem is referenced by:  imanonrel  43713