Theorem resnonrel 44203

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 3969	. . . 4 ⊢ 𝐵 ⊆ V
2		ssres2 6001	. . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)
4		cnvnonrel 44199	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	cnveqi 5858	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅
6		cnvcnv2 6190	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V)
7		cnv0 5867	. . . 4 ⊢ ◡∅ = ∅
8	5, 6, 7	3eqtr3i 2800	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅
9	3, 8	sseqtri 3993	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅
10		ss0b 4364	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅)
11	9, 10	mpbi 233	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Syntax hints:   = wceq 1567  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ^◡ccnv 5658   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-res 5671

This theorem is referenced by:  imanonrel  44204