![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > resnonrel | Structured version Visualization version GIF version |
Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
Ref | Expression |
---|---|
resnonrel | ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3969 | . . . 4 ⊢ 𝐵 ⊆ V | |
2 | ssres2 5966 | . . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V) |
4 | cnvnonrel 41867 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
5 | 4 | cnveqi 5831 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅ |
6 | cnvcnv2 6146 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V) | |
7 | cnv0 6094 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 5, 6, 7 | 3eqtr3i 2773 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅ |
9 | 3, 8 | sseqtri 3981 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ |
10 | ss0b 4358 | . 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅) | |
11 | 9, 10 | mpbi 229 | 1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3446 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 ◡ccnv 5633 ↾ cres 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-res 5646 |
This theorem is referenced by: imanonrel 41872 |
Copyright terms: Public domain | W3C validator |