![]() |
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 4002 | . . . 4 ⊢ 𝐵 ⊆ V | |
2 | ssres2 6007 | . . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V) |
4 | cnvnonrel 42941 | . . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ | |
5 | 4 | cnveqi 5871 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅ |
6 | cnvcnv2 6191 | . . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V) | |
7 | cnv0 6139 | . . . 4 ⊢ ◡∅ = ∅ | |
8 | 5, 6, 7 | 3eqtr3i 2763 | . . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅ |
9 | 3, 8 | sseqtri 4014 | . 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ |
10 | ss0b 4393 | . 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 1534 Vcvv 3469 ∖ cdif 3941 ⊆ wss 3944 ∅c0 4318 ◡ccnv 5671 ↾ cres 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-res 5684 |
This theorem is referenced by: imanonrel 42946 |
Copyright terms: Public domain | W3C validator |